draft-irtf-cfrg-vdaf-18
Internet-Draft
VDAF
January 2026
Barnes, et al.
Expires 3 August 2026
[Page]
Workgroup:
CFRG
Internet-Draft:
draft-irtf-cfrg-vdaf-18
Published:
30 January 2026
Intended Status:
Informational
Expires:
3 August 2026
Authors:
R. L. Barnes
Cisco
D. Cook
ISRG
C. Patton
Cloudflare
P. Schoppmann
Google
Verifiable Distributed Aggregation Functions
Abstract
This document describes Verifiable Distributed Aggregation Functions (VDAFs), a
family of multi-party protocols for computing aggregate statistics over user
measurements. These protocols are designed to ensure that, as long as at least
one aggregation server executes the protocol honestly, individual measurements
are never seen by any server in the clear. At the same time, VDAFs allow the
servers to detect if a malicious or misconfigured client submitted an invalid
measurement. Two concrete VDAFs are specified, one for general-purpose
aggregation (Prio3) and another for heavy hitters (Poplar1).
Discussion Venues
This note is to be removed before publishing as an RFC.
Discussion of this document takes place on the
Crypto Forum Research Group mailing list (cfrg@ietf.org),
which is archived at
Source for this draft and an issue tracker can be found at
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This Internet-Draft will expire on 3 August 2026.
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Copyright (c) 2026 IETF Trust and the persons identified as the
document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal
Provisions Relating to IETF Documents
) in effect on the date of
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respect to this document.
Table of Contents
1.
Introduction
(RFC EDITOR: remove this paragraph.) The source for this draft and the
reference implementation can be found at
The ubiquity of the Internet makes it an ideal platform for measurement of
large-scale phenomena, whether public health trends or the behavior of computer
systems at scale. There is substantial overlap, however, between information
that is valuable to measure and information that users consider private.
For example, consider an application that provides health information to users.
The operator of an application might want to know which parts of their
application are used most often, as a way to guide future development of the
application. Specific users' patterns of usage, though, could reveal sensitive
things about them, such as which users are researching a given health
condition.
In many situations, the measurement collector is only interested in aggregate
statistics, e.g., which portions of an application are most used or what
fraction of people have experienced a given disease. Thus systems that provide
aggregate statistics while protecting individual measurements can deliver the
value of the measurements while protecting users' privacy.
This problem is often formulated in terms of differential privacy (DP)
Dwo06
. Roughly speaking, a data aggregation system that is differentially
private ensures that the degree to which any individual measurement influences
the value of the aggregate result can be precisely controlled. For example, in
systems like RAPPOR
EPK14
, each user samples noise from a well-known
distribution and adds it to their measurement before submitting to the
aggregation server. The aggregation server then adds up the noisy measurements,
and because it knows the distribution from which the noise was sampled, it can
estimate the true sum with reasonable accuracy.
Even when noise is added to the measurements, collecting them in the clear
still reveals a significant amount of information to the collector. On the one
hand, depending on the "amount" of noise a client adds to its measurement, it
may be possible for a curious collector to make a reasonable guess of the
measurement's true value. On the other hand, the more noise the clients add,
the less reliable will be the server's estimate of the aggregate. Thus systems
relying solely on a DP mechanism must strike a delicate balance between privacy
and utility.
Another way of constructing a privacy-preserving measurement system is to use
multi-party computation (MPC). The goal of such a system is that no participant
in the protocol should learn anything about an individual measurement beyond
what it can deduce from the aggregate. MPC achieves this goal by distributing
the computation of the aggregate across multiple aggregation servers, one of
which is presumed to be honest, i.e., not under control of the attacker.
Moreover, MPC can be composed with various DP mechanisms to ensure the
aggregate itself does not leak too much information about any one of the
measurements
MPRV09
This document describes two classes of MPC protocols, each aiming for a
different set of goals.
In a Distributed Aggregation Function (DAF,
Section 4
), each client splits its
measurement into multiple secret shares, one for each aggregation
server. DAFs require two properties of the secret sharing scheme. First, one
can reconstruct the underlying measurement by simply adding up all of the
shares. (Typically the shares are vectors over some finite field.) Second,
given all but one of the shares, it is impossible to learn anything about the
underlying measurement. These properties give rise to a simple strategy for
privately aggregating the measurements: each aggregation server adds up its
measurement shares locally before revealing their sum to the data collector;
then all the data collector has to do is add up these sums to get the aggregate
result.
This strategy is compatible with any aggregation function that can be
represented as the sum of some encoding of the measurements. Examples include:
summary statistics such as sum, mean, and standard deviation; estimation of
quantiles, e.g., median; histograms; linear regression; or counting data
structures, e.g., Bloom filters. However, not all functions fit into this
rubric, as it is constrained to linear computations over the encoded
measurements.
In fact, this document's framework admits DAFs with slightly more
functionality, computing aggregation functions of the form
F(agg_param, meas_1, ..., meas_M) =
G(agg_param, meas_1) + ... + G(agg_param, meas_M)
where
meas_1, ..., meas_M
are the measurements,
is a possibly
non-linear function, and
agg_param
is a parameter of that function chosen by
the data collector. This paradigm, known as function secret sharing
BGI15
allows for more sophisticated data analysis tasks, such as grouping metrics by
private client attributes
MPDST25
or computing heavy hitters
BBCGGI21
(More on the latter task below.)
The second class of protocols defined in this document are called Verifiable
Distributed Aggregation Functions (VDAFs,
Section 5
). In addition to being
private, VDAFs are verifiable in the following sense. By design, a secret
sharing of a valid measurement, e.g., a number between 1 and 10, is
indistinguishable from a secret sharing of an invalid measurement, e.g., a
number larger than 10. This means that DAFs are vulnerable to attacks from
malicious clients attempting to disrupt the computation by submitting invalid
measurements. VDAFs are designed to allow the servers to interact with one
another in order to detect and remove these measurements prior to
aggregation. This document refers to this property as verifiability. (This is
also called robustness in prior work
CGB17
DPRS23
.)
Achieving verifiability using the cryptographic techniques described in this
document requires a significant amount of interaction between the servers. DAFs
on the other hand are non-interactive, making them easier to deploy; but they
do not on their own allow for verifying the validity of the measurements. This
may be tolerable in some applications. For instance, if the client's software
is executed in a trusted execution environment, it may be reasonable to assume
that no client is malicious.
The DAF and VDAF abstractions encompass a variety of MPC techniques in the
literature. These protocols vary in their operational and security
requirements, sometimes in subtle but consequential ways. This document
therefore has two important goals:
Provide higher-level protocols, like
DAP
, with
a simple, uniform interface for accessing privacy-preserving measurement
schemes, document relevant operational and security requirements, and
specify constraints for safe usage:
General patterns of communications among the various actors involved in
the system (clients, aggregation servers, and the collector of the
aggregate result);
Capabilities of a malicious coalition of parties attempting to divulge
information about client measurements; and
Conditions that are necessary to ensure that malicious clients cannot
corrupt the computation.
Provide cryptographers with design criteria that provide a clear deployment
roadmap for new constructions of privacy-preserving measurement systems.
This document also specifies two concrete VDAF schemes, each based on a protocol
from the literature.
The Prio system
CGB17
allows for the privacy-preserving computation of a
variety of aggregate statistics, combining additive secret sharing as
described above with a mechanism for checking the validity of each
measurement.
Section 7
specifies Prio3, a VDAF that follows the same overall
framework as the original Prio protocol, but incorporates techniques
introduced in
BBCGGI19
that result in significant performance gains.
The Poplar protocol
BBCGGI21
solves a problem known as private
heavy-hitters. In this problem, each client holds a bit-string, and the goal
of the aggregation servers is to compute the set of strings that occur at
least
times for some threshold
. The core primitive in their protocol
is a secret sharing of a point function
GI14
(denoted
above) that
allows the servers to privately count how many of the clients' strings begin
with a given prefix (
agg_param
in the notation above).
Section 8
specifies a VDAF called Poplar1 that implements this prefix counting
functionality and describe how it is used in the heavy hitters protocol.
The remainder of this document is organized as follows:
Section 2
lists
definitions and conventions used in the remainder of the document;
Section 3
gives a brief overview of DAFs and VDAFs, the parties involved in the
computation, and the requirements for non-collusion;
Section 4
defines the syntax
for DAFs;
Section 5
defines the syntax for VDAFs;
Section 6
defines various
functionalities that are common to the constructions defined in this document;
Section 7
specifies Prio3;
Section 8
specifies Poplar1; and
Section 9
enumerates security considerations for DAFs and VDAFs in general and the
Prio3 and Poplar1 constructions in particular.
1.1.
Change Log
(RFC EDITOR: remove this section.)
(*) Indicates a change that breaks wire compatibility with the previous draft.
18:
FLP: Represent the gadget polynomials in the Lagrange basis instead of the
monomial basis. This allows us to take advantage of additional algorithmic
improvements described in
Faz25
. (*)
Prio3Sum, Prio3SumVec, Prio3MultihotCountVec: Unify the bit decomposition
used in range proofs. The new technique allows for arbitrary ranges with only
ceil(log2(max_measurement))
field elements. The previous technique required
twice as many. (*)
17:
Rename VDAF preparation to VDAF verification.
Update the reference for TurboSHAKE128 to
RFC9861
Prio3: Acknowledge an optimization pointed out in
Faz25
16:
Align document with guidelines for ASCII-safe mathematical notation from
Section 3.3.1.6
of [
I-D.draft-irtf-cfrg-cryptography-specification-02
Address feedback from Crypto Review Panel
PANEL-FEEDBACK
15:
Simplify the ping-pong API for 2-party preparation by merging the outbound
message into the state object. This reduces the number of cases the caller
has to handle.
Update the test vector format. First, enrich the test vector schema to
express negative test cases. Second, change the encoding of output shares to
match the aggregate shares.
14:
Poplar1: When decoding an aggregation parameter, require the padding bits
after each prefix to be cleared.
13:
(V)DAF: Replace the one-shot aggregation API with a streaming API. Each
Aggregator initializes aggregation, then updates its aggregate share as
output shares are produced. The scheme also specifies a method for merging
multiple aggregate shares.
Poplar1: Move prefix uniqueness and ordering checks from
prep_init()
to
is_valid()
Poplar1: Use
bool
to represent control bits instead of
Field2
Prio3MultihotCountVec: Change the measurement type from
list[int]
to
list[bool]
Security considerations: Define our threat model for side channel attacks and
enumerate the parts of the spec that are most relevant to implementers.
Improve the specification of each Prio3 variant by listing each
implementation of
Valid
and
Gadget
in full. Gadgets are listed in a new
appendix section.
Improve the specification of the FLP system by listing the proof-generation,
query, and decision algorithms in full. The wrapper gadgets are listed in the
new section of the appendix for gadgets.
Add a section with a high-level overview of the IDPF construction.
Move some sections around: move ping-pong and star topologies under
communication patterns for VDAF preparation; move FLP proof generation,
query, and decision algorithms up one level; move privacy considerations for
aggregation parameters up one level; and move safe usage of IDPF outputs up
one level.
12:
(V)DAF: Add an application context string parameter to sharding and
preparation. The motivation for this change is to harden Prio3 against
offline attacks. More generally, however, it allows designing schemes for
which correct execution requires agreement on the application context.
Accordingly, both Prio3 and Poplar1 have been modified to include the context
in the domain separation tag of each XOF invocation. (*)
Prio3: Improve soundness of the base proof system and the circuits of some
variants. Generally speaking, wherever we evaluate a univariate polynomial at
a random point, we can instead evaluate a multivariate polynomial of lower
degree. (*)
Prio3: Replace the helper's measurement and proof share seeds with a single
seed. (*)
Prio3Sum: Update the circuit to support a more general range check and avoid
using joint randomness. (*)
Prio3Histogram, Prio3MultihotCountVec: Move the final reduction of the
intermediate outputs out of the circuit. (*)
IDPF: Add the application context string to key generation and evaluation and
bind it to the fixed AES key. (*)
IDPF: Use XofTurboShake128 for deriving the leaf nodes in order to ensure the
construction is extractable. (*)
IDPF: Simplify the public share encoding. (*)
XofTurboShake128: Change
SEED_SIZE
from 16 bytes to 32 to mitigate offline
attacks on Prio3 robustness. In addition, allow seeds of different lengths so
that we can continue to use XofTurboShake128 with IDPF. (*)
XofTurboShake128, XofFixedKeyAes128: Increase the length prefix for the
domain separation tag from one by to two bytes. This is to accommodate the
application context. (*)
Reassign codepoints for all Prio3 variants and Poplar1. (*)
Security considerations: Add a section on defense-in-depth measures taken by
Prio3 and Poplar1 and more discussion about choosing FLP parameters.
11:
Define message formats for the Poplar1 aggregation parameter and IDPF public
share.
IDPF: Require the IDPF binder must be a random nonce.
VDAF: Replace the pseudocode description of the ping-ping topology with
Python and sketch the star topology.
DAF: Align aggregation parameter validation with VDAF.
Replace
Union[A, B]
type with
A | B
Rename FFT ("Fast Fourier Transform") with NTT ("Number Theoretic
Transform").
10:
Define Prio3MultihotCountVec, a variant of Prio3 for aggregating bit vectors
with bounded weight.
FLP: Allow the output of the circuit to be a vector. This makes it possible
to skip joint randomness derivation in more cases.
Poplar1: On the first round of preparation, handle
None
as an error.
Previously this message was interpreted as a length-3 vector of zeros.
Prio3: Move specification of the field from the FLP validity circuit to the
VDAF itself.
Clarify the extent to which the attacker controls the network in our threat
models for privacy and robustness.
Clean up various aspects of the code, including: Follow existing
object-oriented programming patterns for Python more closely; make the type
hints enforceable; and avoid shadowing variables.
Poplar1: Align terminology with
BBCGGI23
IDPF: Add guidance for encoding byte strings as indices.
09:
Poplar1: Make prefix tree traversal stricter by requiring each node to be a
child of a node that was already visited. This change is intended to make it
harder for a malicious Aggregator to steer traversal towards
non-heavy-hitting measurements.
Prio3: Add more explicit guidance for choosing the field size.
IDPF: Define extractability and clarify (un)safe usage of intermediate prefix
counts. Accordingly, add text ensuring public share consistency to security
considerations.
08:
Poplar1: Bind the report nonce to the authenticator vector programmed into
the IDPF. (*)
IdpfPoplar: Modify
extend()
by stealing each control bit from its
corresponding seed. This improves performance by reducing the number of AES
calls per level from 3 to 2. The cost is a slight reduction in the concrete
privacy bound. (*)
Prio3: Add support for generating and verifying multiple proofs per
measurement. This enables a trade-off between communication cost and runtime:
if more proofs are used, then a smaller field can be used without impacting
robustness. (*)
Replace SHAKE128 with TurboSHAKE128. (*)
07:
Rename PRG to XOF ("eXtendable Output Function"). Accordingly, rename PrgSha3
to XofShake128 and PrgFixedKeyAes128 to XofFixedKeyAes128. "PRG" is a misnomer
since we don't actually treat this object as a pseudorandom generator in
existing security analysis.
Replace cSHAKE128 with SHAKE128, re-implementing domain separation for the
customization string using a simpler scheme. This change addresses the
reality that implementations of cSHAKE128 are less common. (*)
Define a new VDAF, called Prio3SumVec, that generalizes Prio3Sum to a vector
of summands.
Prio3Histogram: Update the codepoint and use the parallel sum optimization
introduced by Prio3SumVec to reduce the proof size. (*)
Daf, Vdaf: Rename interface methods to match verbiage in the draft.
Daf: Align with Vdaf by adding a nonce to
shard()
and
prep()
Vdaf: Have
prep_init()
compute the first prep share. This change is
intended to simplify the interface by making the input to
prep_next()
not
optional.
Prio3: Split sharding into two auxiliary functions, one for sharding with
joint randomness and another without. This change is intended to improve
readability.
Fix bugs in the ping-pong interface discovered after implementing it.
06:
Vdaf: Define a wrapper interface for preparation that is suitable for the
"ping-pong" topology in which two Aggregators exchange messages over a
request/response protocol, like HTTP, and take turns executing the
computation until input from the peer is required.
Prio3Histogram: Generalize the measurement type so that the histogram can be
used more easily with discrete domains. (*)
Daf, Vdaf: Change the aggregation parameter validation algorithm to take the
set of previous parameters rather than a list. (The order of the parameters
is irrelevant.)
Daf, Vdaf, Idpf: Add parameter
RAND_SIZE
that specifies the number of
random bytes consumed by the randomized algorithm (
shard()
for Daf and Vdaf
and
gen()
for Idpf).
05:
IdpfPoplar: Replace PrgSha3 with PrgFixedKeyAes128, a fixed-key mode for
AES-128 based on a construction from
GKWWY20
. This change is intended to
improve performance of IDPF evaluation. Note that the new PRG is not suitable
for all applications. (*)
Idpf: Add a binder string to the key-generation and evaluation algorithms.
This is used to plumb the nonce generated by the Client to the PRG.
Plumb random coins through the interface of randomized algorithms.
Specifically, add a random input to (V)DAF sharding algorithm and IDPF
key-generation algorithm and require implementations to specify the length of
the random input. Accordingly, update Prio3, Poplar1, and IdpfPoplar to match
the new interface. This change is intended to improve coverage of test
vectors.
Use little-endian byte-order for field element encoding. (*)
Poplar1: Move the last step of sketch evaluation from
prep_next()
to
prep_shares_to_prep()
04:
Align security considerations with the security analysis of
DPRS23
Vdaf: Pass the nonce to the sharding algorithm.
Vdaf: Rather than allow the application to choose the nonce length, have each
implementation of the Vdaf interface specify the expected nonce length. (*)
Prg: Split "info string" into two components: the "customization string",
intended for domain separation; and the "binder string", used to bind the
output to ephemeral values, like the nonce, associated with execution of a
(V)DAF.
Replace PrgAes128 with PrgSha3, an implementation of the Prg interface based
on SHA-3, and use the new scheme as the default. Accordingly, replace
Prio3Aes128Count with Prio3Count, Poplar1Aes128 with Poplar1, and so on. SHA-3
is a safer choice for instantiating a random oracle, which is used in the
analysis of Prio3 of
DPRS23
. (*)
Prio3, Poplar1: Ensure each invocation of the Prg uses a distinct
customization string, as suggested by
DPRS23
. This is intended to make
domain separation clearer, thereby simplifying security analysis. (*)
Prio3: Replace "joint randomness hints" sent in each input share with "joint
randomness parts" sent in the public share. This reduces communication
overhead when the number of shares exceeds two. (*)
Prio3: Bind nonce to joint randomness parts. This is intended to address
birthday attacks on robustness pointed out by
DPRS23
. (*)
Poplar1: Use different Prg invocations for producing the correlated randomness
for inner and leaf nodes of the IDPF tree. This is intended to simplify
implementations. (*)
Poplar1: Don't bind the candidate prefixes to the verifier randomness. This is
intended to improve performance, while not impacting security. According to
the analysis of
DPRS23
, it is necessary to restrict Poplar1 usage such
that no report is aggregated more than once at a given level of the IDPF tree;
otherwise, attacks on privacy may be possible. In light of this restriction,
there is no added benefit of binding to the prefixes themselves. (*)
Poplar1: During preparation, assert that all candidate prefixes are unique
and appear in order. Uniqueness is required to avoid erroneously rejecting a
valid report; the ordering constraint ensures the uniqueness check can be
performed efficiently. (*)
Poplar1: Increase the maximum candidate prefix count in the encoding of the
aggregation parameter. (*)
Poplar1: Bind the nonce to the correlated randomness derivation. This is
intended to provide defense-in-depth by ensuring the Aggregators reject the
report if the nonce does not match what the Client used for sharding. (*)
Poplar1: Clarify that the aggregation parameter encoding is
OPTIONAL
Accordingly, update implementation considerations around cross-aggregation
state.
IdpfPoplar: Add implementation considerations around branching on the values
of control bits.
IdpfPoplar: When decoding the control bits in the public share, assert
that the trailing bits of the final byte are all zero. (*)
03:
Define codepoints for (V)DAFs and use them for domain separation in Prio3 and
Poplar1. (*)
Prio3: Align joint randomness computation with revised paper
BBCGGI19
This change mitigates an attack on robustness. (*)
Prio3: Remove an intermediate PRG evaluation from query randomness generation.
(*)
Add additional guidance for choosing FFT-friendly fields.
02:
Complete the initial specification of Poplar1.
Extend (V)DAF syntax to include a "public share" output by the Client and
distributed to all of the Aggregators. This is to accommodate "extractable"
IDPFs as required for Poplar1. (See
BBCGGI21
, Section 4.3 for details.)
Extend (V)DAF syntax to allow the unsharding step to take into account the
number of measurements aggregated.
Extend FLP syntax by adding a method for decoding the aggregate result from a
vector of field elements. The new method takes into account the number of
measurements.
Prio3: Align aggregate result computation with updated FLP syntax.
Prg: Add a method for statefully generating a vector of field elements.
Field: Require that field elements are fully reduced before decoding. (*)
Define new field Field255.
01:
Require that VDAFs specify serialization of aggregate shares.
Define Distributed Aggregation Functions (DAFs).
Prio3: Move proof verifier check from
prep_next()
to
prep_shares_to_prep()
. (*)
Remove public parameter and replace verification parameter with a
"verification key" and "Aggregator ID".
2.
Conventions
The key words "
MUST
", "
MUST NOT
", "
REQUIRED
", "
SHALL
", "
SHALL NOT
", "
SHOULD
", "
SHOULD NOT
", "
RECOMMENDED
", "
NOT RECOMMENDED
",
MAY
", and "
OPTIONAL
" in this document are to be interpreted as
described in BCP 14
RFC2119
RFC8174
when, and only when, they
appear in all capitals, as shown here.
Algorithms in this document are written in Python (compatible with Python 3.12
or later). A fatal error in a program (e.g., failure to parse one of the
function parameters) is usually handled by raising an exception.
In Python, array indexing starts with
, e.g.,
x[0]
is the first element
and
x[len(x)-1]
is the last of
. It is also possible to index from the end
of the list, e.g.,
x[-1]
is the last element of
Python uses the symbols
, and
as binary operators. When the
operands are integers, these have the usual meaning, except:
Division results in a floating point number. Python includes a similar
operator,
x // y
, which is short for
floor(x / y)
When
and
are byte strings,
x + y
denotes their concatenation, i.e.,
concat(x, y)
as defined below.
Finite field arithmetic overloads these operators; see
Section 6.1
Exponentiation is denoted by
x ** y
in Python.
Bitwise exclusive or is denoted by
x ^ y
in Python.
Type hints are used to define input and output types:
The type variable
is used in signatures to signify any type that is a
subclass of
Field
Section 6.1
).
bytes
is a byte string.
int
is an integer.
Generic
is used in class definitions to explicitly declare type variables
of generic classes.
Any
is the universal supertype, which admits values of any type.
Optional[T]
is shorthand for
T | None
. Its value may be
None
or have
type
Self
represents the containing class of the method definition in which it
appears.
Sequence[T]
is either a list or tuple of values of type
This document defines several byte-string constants. When comprised of printable
ASCII characters, they are written as Python 3 byte-string literals (e.g.,
b'some constant string'
).
A global constant
VERSION
of type
int
is defined, which algorithms are free
to use as desired. Its value
SHALL
be
18
This document describes algorithms for multi-party computations in which the
parties typically communicate over a network. Wherever a quantity is defined
that must be transmitted from one party to another, this document prescribes
a particular encoding of that quantity as a byte string.
Some common functionalities:
additive_secret_share(x: list[F], num_shares: int, field: type[F]) ->
list[list[F]]
takes a vector
of field elements and returns
num_shares
vectors of length
len(x)
such that they all add up to the input vector.
Note that this function is not used normatively in this document.
byte(x: int) -> bytes
returns the representation of the integer
in the
range
[0, 256)
as a single-byte byte string.
cast(typ: type[T], x: object) -> T
returns the input value unchanged.
This is only present to assist with static analysis of the Python code.
Type checkers will ignore the inferred type of the input value, and assume
the output value has the given type.
concat(parts: list[bytes]) -> bytes
returns the concatenation of the input
byte strings, i.e.,
parts[0] + ... + parts[len(parts)-1]
from_be_bytes(encoded: bytes) -> int
decodes a big-endian byte string,
i.e., returns the integer
for which
to_be_bytes(x, len(encoded)) ==
encoded
from_le_bytes(encoded: bytes) -> int
decodes a little-endian byte string,
i.e., returns the integer
for which
to_le_bytes(x, len(encoded)) ==
encoded
front(len: int, x: list[Any]) -> tuple[list[Any], list[Any]]
splits
into two vectors, where the first vector is made up of the first
len
elements of
and the second is made up of the remaining elements. This
function is equivalent to
(x[:len], x[len:])
gen_rand(len: int) -> bytes
returns a byte array of the requested length
len
) generated by a cryptographically secure pseudorandom number generator
(CSPRNG).
next_power_of_2(x: int) -> int
returns the smallest integer
greater than or equal to
that is also a power of two.
assert_power_of_2(x: int) -> int
returns
math.ceil(math.log2(n))
if
is a positive power of two and raises an exception otherwise.
range(stop: int)
or
range(start: int, stop: int[, step: int])
is the range
function from the Python standard library. The one-argument form returns the
integers from zero (inclusive) to
stop
(exclusive). The two- and
three-argument forms allow overriding the start of the range and overriding
the step between successive output values.
to_be_bytes(x: int, len: int) -> bytes
converts an integer
whose value
is in the range
[0, 2**(8*len))
to a big-endian byte string of length
len
to_le_bytes(x: int, len: int) -> bytes
converts an integer
whose value
is in the range
[0, 2**(8*len))
to a little-endian byte string of length
len
xor(left: bytes, right: bytes) -> bytes
returns the bitwise XOR of
left
and
right
. An exception is raised if the inputs are not the same length.
zeros(len: int) -> bytes
returns an array of bytes of the requested
length (
len
). Each element of the array is set to zero.
2.1.
Mathematical Notation
The following symbols are used in mathematical notation.
Table 1
Mathematical Operators and Symbols
ASCII glyph(s)
Description
Comment
+, -
Addition / subtraction
Needs to be constant time. Used for addition and subtraction of field elements as well as numbers.
Addition / subtraction
Needs to be constant time. Used for multiplication of field elements as well as numbers.
**
Exponentiation
Needs to be constant time. Used for exponentiation of field elements as well as numbers.
Division
Needs to be constant time. Used for division of field elements (i.e. multiplication by inverse) as well as numbers.
||
Concatenation
Used for concatenation of byte strings and bit strings.
3.
Overview
Figure 1
Overall data flow of a (V)DAF.
There are three types of actors in a DAF- or VDAF-based private measurement
system: Clients, Aggregators, and the Collector. The overall flow of the
measurement process is illustrated in
Figure 1
. The steps are as
follows:
To submit an individual measurement, a Client shards its measurement into
"input shares" and sends one input share to each Aggregator. This document
sometimes refers to this sequence of input shares collectively as the
Client's "report". (The report contains a few more items needed to process
the measurement; these are described in
Section 4
.)
Once an Aggregator receives an input share from each Client, it processes
the input shares into a value called an "aggregate share" and sends it the
Collector. The aggregate share is a secret share of the aggregate
representation of the measurements.
Once the Collector has received an aggregate share from each Aggregator, it
combines them into the aggregate representation of the measurements, called
the "aggregate result".
For DAFs, this second step involves a process called "preparation" in which the
Aggregator refines each input share into an intermediate representation called
an "output share". The output shares are then combined into the aggregate share
as shown in
Figure 2
Figure 2
DAF preparation of input shares into output shares and aggregation of output shares into an aggregate share. Executed by each Aggregator. M denotes the number of measurements being aggregated.
In the case of VDAFs (
Section 5
), this process is instead called "verification"
and involves interacting with the other Aggregators. It will fail if the
underlying measurement is invalid, in which case the report is rejected and not
included in the aggregate result.
Aggregators are a new class of actor relative to traditional measurement systems
where Clients submit measurements to a single server. They are critical for
both the privacy properties of the system and, in the case of VDAFs, the
validity of the aggregate results obtained. The privacy properties of the
system are assured by non-collusion among Aggregators, and Aggregators are the
entities that perform validation of Client measurements. Thus Clients trust
Aggregators not to collude (typically it is required that at least one
Aggregator is honest; see
Section 9.8
), and Collectors trust
Aggregators to correctly run the protocol.
Within the bounds of the non-collusion requirements of a given (V)DAF instance,
it is possible for the same entity to play more than one role. For example, the
Collector could also act as an Aggregator, effectively using the other
Aggregator(s) to augment a basic client-server protocol.
This document describes the computations performed by the actors in this
system. It is up to the higher-level protocol making use of the (V)DAF to
arrange for the required information to be delivered to the proper actors in
the proper sequence. In general, it is assumed that all communications are
confidential and mutually authenticated, with the exception that Clients
submitting measurements may be anonymous.
4.
Definition of DAFs
By way of a gentle introduction to VDAFs, this section describes a simpler class
of schemes called Distributed Aggregation Functions (DAFs). Unlike VDAFs, DAFs
do not provide verifiability of the computation. Clients must therefore be
trusted to compute their input shares correctly. Because of this fact, the use
of a DAF is
NOT RECOMMENDED
for most applications. See
Section 9
for
additional discussion.
A DAF scheme is used to compute a particular "aggregation function" over a set
of measurements generated by Clients. Depending on the aggregation function, the
Collector might select an "aggregation parameter" and disseminate it to the
Aggregators. The semantics of this parameter is specific to the aggregation
function, but in general it is used to represent the set of "queries" that can
be made by the Collector on the batch of measurements. For example, the
aggregation parameter is used to represent the prefixes in the prefix-counting
functionality of of Poplar1 discussed in
Section 1
Execution of a DAF has four distinct stages:
Sharding: Each Client generates input shares from its measurement and
distributes them among the Aggregators. In addition to the input shares, the
client generates a "public share" during this step that is disseminated to
all of the Aggregators.
Preparation: Each Aggregator converts each input share into an output share
compatible with the aggregation function. This computation involves the
aggregation parameter. In general, each aggregation parameter may result in a
different output share.
Aggregation: Each Aggregator combines a sequence of output shares into its
aggregate share and sends the aggregate share to the Collector.
Unsharding: The Collector combines the aggregate shares into the aggregate
result.
Sharding and preparation are done once per measurement. Aggregation and
unsharding are done over a batch of measurements (more precisely, over the
recovered output shares).
A concrete DAF specifies the algorithm for the computation needed in each of
these stages. The interface, denoted
Daf
, is defined in the remainder of this
section. In addition, a concrete DAF defines the associated constants and types
enumerated in the following table.
Table 2
Constants and types defined by each concrete DAF.
Parameter
Description
ID: int
Algorithm identifier for this DAF, in the range
[0, 2**32)
SHARES: int
Number of input shares into which each measurement is sharded.
NONCE_SIZE: int
Size of the nonce associated with each report.
RAND_SIZE: int
Size of each random byte string consumed by the sharding algorithm.
Measurement
Type of each measurement.
PublicShare
Type of each public share.
InputShare
Type of each input share.
AggParam
Type of the aggregation parameter.
OutShare
Type of each output share.
AggShare
Type of each aggregate share.
AggResult
Type of the aggregate result.
The types in this table define the inputs and outputs of DAF methods at various
stages of the computation. Some of these values need to be written to the
network in order to carry out the computation. In particular, it is
RECOMMENDED
that concrete instantiations of the
Daf
interface specify a standard encoding
for the
PublicShare
InputShare
AggParam
, and
AggShare
types.
4.1.
Sharding
Figure 3
Illustration of the sharding algorithm.
The sharding algorithm run by each Client is defined as follows:
daf.shard(ctx: bytes, measurement: Measurement, nonce: bytes, rand: bytes)
-> tuple[PublicShare, list[InputShare]]
consumes the "application context"
(defined below), a measurement, and a nonce and produces the public share,
distributed to each of the Aggregators, and the input shares, one for each
Aggregator.
Pre-conditions:
nonce
MUST
have length equal to
daf.NONCE_SIZE
and
MUST
be generated
using a cryptographically secure random number generator (CSPRNG).
rand
consists of the random bytes consumed by the algorithm. It
MUST
have
length equal to
daf.RAND_SIZE
and
MUST
be generated using a CSPRNG.
Post-conditions:
The number of input shares
MUST
equal
daf.SHARES
Sharding is bound to a specific "application context". The application context
is a string intended to uniquely identify an instance of the higher level
protocol that uses the DAF. The goal of binding the application to DAF
execution is to ensure that aggregation succeeds only if the Clients and
Aggregators agree on the application context. (Preparation binds the
application context, too; see
Section 4.2
.) Note that, unlike VDAFs
Section 5
), there is no explicit signal of disagreement; it may only manifest
as a garbled aggregate result.
The nonce is a public random value associated with the report. It is referred
to as a nonce because normally it will also be used as a unique identifier for
that report in the context of some application. The randomness requirement is
especially important for VDAFs, where it may be used by the Aggregators to
derive per-report randomness for verification of the computation. See
Section 9.2
for details.
4.2.
Preparation
Figure 4
Illustration of preparation.
Once an Aggregator has received the public share and its input share, the next
step is to prepare the input share for aggregation. This is accomplished using
the preparation algorithm:
daf.prep(ctx: bytes, agg_id: int, agg_param: AggParam, nonce: bytes,
public_share: PublicShare, input_share: InputShare) -> OutShare
consumes the
public share and one of the input shares generated by the Client, the
application context, the Aggregator's unique identifier, the aggregation
parameter selected by the Collector, and the report nonce and returns an
output share.
Pre-conditions:
agg_id
MUST
be in the range
[0, daf.SHARES)
and match the index of
input_share
in the sequence of input shares produced by the Client.
nonce
MUST
have length
daf.NONCE_SIZE
The Aggregators
MUST
agree on the value of the aggregation parameter.
Otherwise, the aggregate result may be computed incorrectly by the Collector.
4.3.
Validity of Aggregation Parameters
In general, it is permissible to aggregate a batch of reports multiple times.
However, to prevent privacy violations, DAFs may impose certain restrictions on
the aggregation parameters selected by the Collector. Restrictions are
expressed by the aggregation parameter validity function:
daf.is_valid(agg_param: AggParam, previous_agg_params: list[AggParam]) ->
bool
returns
True
if
agg_param
is allowed given the sequence
previous_agg_params
of previously accepted aggregation parameters.
Prior to accepting an aggregation parameter from the Collector and beginning
preparation, each Aggregator
MUST
validate it using this function.
4.4.
Aggregation
Figure 5
Illustration of aggregation. The number of measurements in the batch is denoted by M.
Once an Aggregator holds an output share, it adds it into its aggregate share
for the batch. This streaming aggregation process is implemented by the
following pair of algorithms:
daf.agg_init(agg_param: AggParam) -> AggShare
returns an empty aggregate
share. It is called to initialize aggregation of a batch of measurements.
daf.agg_update(agg_param: AggParam, agg_share: AggShare, out_share:
OutShare) -> AggShare
accumulates an output share into an aggregate share
and returns the updated aggregate share.
In many situations it is desirable to split an aggregate share across multiple
storage elements, then merge the aggregate shares together just before
releasing the completed aggregate share to the Collector. DAFs facilitate this
with the following method:
daf.merge(agg_param: AggParam, agg_shares: list[AggShare]) -> AggShare
merges a sequence of aggregate shares into a single aggregate share.
4.4.1.
Aggregation Order
For most DAFs and VDAFs, the outcome of aggregation is not sensitive to the
order in which output shares are aggregated. This means that aggregate shares
can be updated or merged with other aggregate shares in any order. For
instance, for both Prio3 (
Section 7
) and Poplar1 (
Section 8
), the aggregate
shares and output shares both have the same type, a vector over some finite
field (
Section 6.1
); and aggregation involves simply adding vectors together.
In theory, however, there may be a DAF or VDAF for which correct execution
requires each Aggregator to aggregate output shares in the same order.
4.5.
Unsharding
Figure 6
Illustration of unsharding.
After the Aggregators have aggregated all measurements in the batch, each sends
its aggregate share to the Collector, who runs the unsharding algorithm to
recover the aggregate result:
daf.unshard(agg_param: AggParam, agg_shares: list[AggShare],
num_measurements: int) -> AggResult
consumes the aggregate shares
and produces the aggregate result.
Pre-conditions:
The length of
agg_shares
MUST
be
SHARES
num_measurements
MUST
equal the number of measurements in the batch.
4.6.
Execution of a DAF
Secure execution of a DAF involves simulating the following procedure over an
insecure network.
def run_daf(
daf: Daf[
Measurement,
AggParam,
PublicShare,
InputShare,
OutShare,
AggShare,
AggResult,
],
ctx: bytes,
agg_param: AggParam,
measurements: list[Measurement]) -> AggResult:
agg_shares: list[AggShare]
agg_shares = [daf.agg_init(agg_param)
for _ in range(daf.SHARES)]
for measurement in measurements:
# Sharding
nonce = gen_rand(daf.NONCE_SIZE)
rand = gen_rand(daf.RAND_SIZE)
(public_share, input_shares) = \
daf.shard(ctx, measurement, nonce, rand)
# Preparation, aggregation
for j in range(daf.SHARES):
out_share = daf.prep(ctx, j, agg_param, nonce,
public_share, input_shares[j])
agg_shares[j] = daf.agg_update(agg_param,
agg_shares[j],
out_share)
# Unsharding
num_measurements = len(measurements)
agg_result = daf.unshard(agg_param, agg_shares,
num_measurements)
return agg_result
The inputs to this procedure include the parameters of the aggregation function
computed by the DAF: an aggregation parameter and a sequence of measurements.
They also include the application context. The procedure prescribes how a DAF is
executed in a "benign" environment in which there is no adversary and the
messages are passed among the protocol participants over secure point-to-point
channels. In reality, these channels need to be instantiated by some "wrapper
protocol", such as
DAP
, that realizes these channels using suitable
cryptographic mechanisms. Moreover, some fraction of the Aggregators (or
Clients) may be malicious and diverge from their prescribed behaviors.
Section 9
describes the execution of the DAF in various adversarial
environments and what properties the wrapper protocol needs to provide in each.
5.
Definition of VDAFs
VDAFs are identical to DAFs except that the non-interactive preparation process
is replaced by an interactive process called "verification" in which, in
addition to refining their input shares into output shares, the Aggregators
also verify that they hold valid output shares.
Failure manifests as an exception raised by one of the algorithms defined in
this section. If an exception is raised during verification, the Aggregators
MUST
remove the report from the batch and not attempt to aggregate it.
Otherwise, a malicious Client can cause the Collector to compute a malformed
aggregate result.
The remainder of this section defines the VDAF interface, denoted by
Vdaf
The attributes listed in
Table 3
are defined by each concrete VDAF.
Table 3
Constants and types defined by each concrete VDAF.
Parameter
Description
ID: int
Algorithm identifier for this VDAF, in the range
[0, 2**32)
SHARES: int
Number of input shares into which each measurement is sharded.
ROUNDS: int
Number of rounds of communication during verification.
NONCE_SIZE: int
Size of each report nonce.
RAND_SIZE: int
Size of each random byte string consumed during sharding.
VERIFY_KEY_SIZE: int
Size of the verification key used during verification.
Measurement
Type of each measurement.
PublicShare
Type of each public share.
InputShare
Type of each input share.
AggParam
Type of the aggregation parameter.
OutShare
Type of each output share.
AggShare
Type of each aggregate share.
AggResult
Type of the aggregate result.
VerifyState
Type of each verification state.
VerifierShare
Type of each verifier share.
VerifierMessage
Type of each verifier message.
Some of the types in the table above need to be written to the network in
order to carry out the computation. It is
RECOMMENDED
that concrete
instantiations of the
Vdaf
interface specify a method of encoding the
PublicShare
InputShare
AggParam
AggShare
VerifierShare
, and
VerifierMessage
types.
Each VDAF is identified by a unique 32-bit integer, denoted
ID
. Identifiers
for each VDAF specified in this document are defined in
Table 19
. The
following method is used by both Prio3 and Poplar1:
def domain_separation_tag(self, usage: int, ctx: bytes) -> bytes:
"""
Format domain separation tag for this VDAF with the given
application context and usage.
Pre-conditions:
- `usage` in the range `[0, 2**16)`
"""
return format_dst(0, self.ID, usage) + ctx
The output, called the "domain separation tag", is used in the constructions
for domain separation. Function
format_dst()
is defined in
Section 6.2.3
5.1.
Sharding
Sharding is as described for DAFs in
Section 4.1
. The public share and
input shares encode additional information used during verification to validate
the output shares before they are aggregated (e.g., the "proof shares" in
Section 7
).
Like DAFs, sharding is bound to the application context via the application
context string. Again, this is intended to ensure that aggregation succeeds
only if the Clients and Aggregators agree on the application context. Unlike
DAFs, however, disagreement on the context should manifest as a verification
failure, causing the report to be rejected without garbling the aggregate
result. The application context also provides some defense-in-depth against
cross protocol attacks; see
Section 9.9
5.2.
Verification
Figure 7
Illustration of interactive VDAF verification.
Verification is organized into a number of rounds. The number of rounds depends
on the VDAF: Prio3 (
Section 7
) has one round and Poplar1 (
Section 8
) has two.
Aggregators retain some local state between successive rounds of verification.
This is referred to as "verification state", usually written as
verify_state
for short.
During each round, each Aggregator broadcasts a message called a "verifier
share", usually written
verifier_share
. The verifier shares are then combined
into a single message called the "verifier message", or
verifier_message
. The
verifier message is computed from public information and therefore
MAY
be
computed by any one of the Aggregators.
The verifier message is disseminated to each of the Aggregators to begin the
next round. An Aggregator begins the first round with its input share and it
begins each subsequent round with the current verification state and the
previous verifier message. Its output in the last round is its output share and
its output in each of the preceding rounds is a verifier share.
Just as for DAF preparation involves an aggregation parameter
Section 4.2
), so does VDAF verification. The aggregation parameter is
consumed by each Aggregator before the first round of communication.
In addition, VDAF verification involves a secret "verification key" held by
each of the Aggregators. This key is used to verify validity of the output
shares they compute. It is up to the high level protocol in which the VDAF is
used to arrange for the distribution of the verification key prior to
generating and processing reports. See
Section 9
for details.
Verification is implemented by the following set of algorithms:
vdaf.verify_init(verify_key: bytes, ctx: bytes, agg_id: int, agg_param:
AggParam, nonce: bytes, public_share: PublicShare, input_share: InputShare)
-> tuple[VerifyState, VerifierShare]
is the deterministic verification state
initialization algorithm run by each Aggregator. It consumes the shared
verification key, the application context, the Aggregator's unique
identifier, the aggregation parameter chosen by the Collector, the report
nonce, the public share, and one of the input shares generated by the Client.
It produces the Aggregator's initial verification state and outbound verifier
share.
Protocols
MUST
ensure that public share consumed by each of the Aggregators is
identical. This is security critical for VDAFs such as Poplar1.
Pre-conditions:
verify_key
MUST
have length
vdaf.VERIFY_KEY_SIZE
agg_id
MUST
be the integer in the range
[0, vdaf.SHARES)
that matches
the index of
input_share
in the sequence of input shares output by the
Client.
nonce
MUST
have length
vdaf.NONCE_SIZE
vdaf.verifier_shares_to_message(ctx: bytes, agg_param: AggParam,
verifier_shares: list[VerifierShare]) -> VerifierMessage
is the
deterministic verifier message pre-processing algorithm. It combines the
verifier shares produced by the Aggregators in the previous round into the
verifier message consumed by each Aggregator to start the next round.
vdaf.verify_next(ctx: bytes, verify_state: VerifyState, verifier_message:
VerifierMessage) -> tuple[VerifyState, VerifierShare] | OutShare
is the
deterministic verification state update algorithm run by each Aggregator. It
updates the Aggregator's verification state (
verify_state
) and returns
either its next verification state and verifier share for the next round or,
if this is the last round, its output share.
An exception may be raised by one of these algorithms, in which case the report
MUST
be deemed invalid and not processed any further.
Implementation note: The verification process accomplishes two tasks: recovery
of output shares from the input shares and ensuring that the recovered output
shares are valid. The abstraction boundary is drawn so that an Aggregator only
recovers an output share if the underlying data is deemed valid (at least,
based on the Aggregator's view of the protocol). Another way to draw this
boundary would be to have the Aggregators recover output shares first, then
verify that they are valid. However, this would allow the possibility of
misusing the API by, say, aggregating an invalid output share. Moreover, in
protocols like Prio+
AGJOP21
based on oblivious transfer, it is necessary
for the Aggregators to interact in order to recover aggregatable output shares
at all.
5.3.
Validity of Aggregation Parameters
Aggregation parameter validation is as described for DAFs in
Section 4.3
. Again, each Aggregator
MUST
validate each
aggregation parameter received from the Collector before beginning verification
with that parameter.
5.4.
Aggregation
Aggregation is identical to DAF aggregation as described in
Section 4.4
. As with DAFs, computation of the VDAF aggregate is not
usually sensitive to the order in which output shares are aggregated. See
Section 4.4.1
5.5.
Unsharding
Unsharding is identical to DAF unsharding as described in
Section 4.5
5.6.
Execution of a VDAF
The following function describes the sequence of computations that are carried
out during VDAF execution:
def run_vdaf(
vdaf: Vdaf[
Measurement,
AggParam,
PublicShare,
InputShare,
OutShare,
AggShare,
AggResult,
VerifyState,
VerifierShare,
VerifierMessage,
],
verify_key: bytes,
agg_param: AggParam,
ctx: bytes,
measurements: list[Measurement]) -> AggResult:
"""
Execute the VDAF for the given measurements, aggregation
parameter (`agg_param`), application context (`ctx`), and
verification key (`verify_key`).
"""
agg_shares = [vdaf.agg_init(agg_param)
for _ in range(vdaf.SHARES)]
for measurement in measurements:
# Sharding: The Client shards its measurement into a report
# consisting of a public share and a sequence of input
# shares.
nonce = gen_rand(vdaf.NONCE_SIZE)
rand = gen_rand(vdaf.RAND_SIZE)
(public_share, input_shares) = \
vdaf.shard(ctx, measurement, nonce, rand)
# Initialize verification: Each Aggregator receives its
# report share (the public share and its input share) from
# the Client and initializes verification.
verify_states = []
outbound_verifier_shares = []
for j in range(vdaf.SHARES):
(state, share) = vdaf.verify_init(verify_key, ctx, j,
agg_param,
nonce,
public_share,
input_shares[j])
verify_states.append(state)
outbound_verifier_shares.append(share)
# Complete verification: The Aggregators execute each round
# of verification until each computes an output share. A
# round begins by gathering the verifier shares and combining
# them into the verifier message. The round ends when each
# uses the verifier message to transition to the next state.
for i in range(vdaf.ROUNDS - 1):
verifier_message = vdaf.verifier_shares_to_message(
ctx, agg_param, outbound_verifier_shares)
outbound_verifier_shares = []
for j in range(vdaf.SHARES):
out = vdaf.verify_next(ctx,
verify_states[j],
verifier_message)
assert isinstance(out, tuple)
(verify_states[j], verifier_share) = out
outbound_verifier_shares.append(verifier_share)
verifier_message = vdaf.verifier_shares_to_message(
ctx, agg_param, outbound_verifier_shares)
# Aggregation: Each Aggregator updates its aggregate share
# with its output share.
for j in range(vdaf.SHARES):
out_share = vdaf.verify_next(
ctx, verify_states[j], verifier_message)
assert not isinstance(out_share, tuple)
agg_shares[j] = vdaf.agg_update(agg_param,
agg_shares[j],
out_share)
# Unsharding: The Collector receives the aggregate shares from
# the Aggregators and combines them into the aggregate result.
num_measurements = len(measurements)
agg_result = vdaf.unshard(agg_param, agg_shares,
num_measurements)
return agg_result
Depending on the VDAF, verification, aggregation, and collection may be carried
out multiple times on the same sequence of reports.
In practice, VDAF execution is distributed across Clients, Aggregators, and
Collectors that exchange messages (i.e., report shares, verifier shares, and
aggregate shares) over an insecure network. The application must therefore take
some additional steps in order to securely execute the VDAF in this
environment. See
Section 9
for details.
5.7.
Communication Patterns for Verification
The only stage of VDAF execution that requires interaction is verification
Section 5.2
). There are a number of ways to coordinate this
interaction; the best strategy depends largely on the number of Aggregators
(i.e.,
vdaf.SHARES
). This section describes two strategies, one specialized
for two Aggregators and another that is suitable for any number of Aggregators.
In each round of verification, each Aggregator writes a verifier share to some
broadcast channel, which is then processed into the verifier message using the
public
verifier_shares_to_message()
algorithm and broadcast to the
Aggregators to start the next round. The goal of this section is to realize
this broadcast channel.
The state machine of each Aggregator is shown below.
Figure 8
State machine of VDAF verification.
State transitions are made when the state is acted upon by the Aggregator's
local inputs and/or messages sent by its co-Aggregators. The initial state is
Start
. The terminal states are:
Rejected
, indicating that the report cannot
be processed any further;
Finished(out_share)
, indicating that the
Aggregator has recovered an output share
out_share
; and
FinishedWithOutbound(out_share, outbound)
, indicating that the Aggregator has
recovered an output share, and has one more outbound message to send. For
completeness, these states are defined in
Appendix B
The methods described in this section are defined in terms of opaque byte
strings. A compatible
Vdaf
MUST
specify methods for encoding public shares,
input shares, verifier shares, verifier messages, and aggregation parameters.
Implementations of Prio3 and Poplar1
MUST
use the encoding schemes specified in
Section 7.2.7
and
Section 8.2.6
respectively.
5.7.1.
The Ping-Pong Topology (Only Two Aggregators)
For VDAFs with precisely two Aggregators (i.e.,
vdaf.SHARES == 2
), the
following "ping pong" communication pattern can be used. It is compatible with
any request/response transport protocol, such as HTTP.
In this section, the initiating Aggregator is called the Leader and the
responding Aggregator is called the Helper. The high-level idea is that the
Leader and Helper will take turns running the computation locally until input
from their peer is required:
For a 1-round VDAF (e.g., Prio3 in
Section 7
), the Leader sends its verifier
share to the Helper, who computes the verifier message locally, computes its
output share, then sends the verifier message to the Leader. Verification
requires just one round trip between the Leader and the Helper.
For a 2-round VDAF (e.g., Poplar1 in
Section 8
), the Leader sends its
first-round verifier share to the Helper, who replies with the first-round
verifier message and its second-round verifier share. In the next request,
the Leader computes its second-round verifier share locally, computes its
output share, and sends the second-round verifier message to the Helper.
Finally, the Helper computes its own output share.
In general, each request includes the Leader's verifier share for the
previous round and/or the verifier message for the current round;
correspondingly, each response consists of the verifier message for the
current round and the Helper's verifier share for the next round.
The Aggregators proceed in this ping-ponging fashion until a step of the
computation fails (indicating the report is invalid and should be rejected) or
verification is completed. All told there are
ceil((vdaf.ROUNDS+1)/2)
requests sent.
Protocol messages are specified in the presentation language of TLS; see
Section 3
of [
RFC8446
. Each message is structured as follows:
enum {
initialize(0),
continue(1),
finish(2),
(255)
} MessageType;
struct {
MessageType type;
select (Message.type) {
case initialize:
opaque verifier_share<0..4294967295>;
case continue:
opaque verifier_message<0..4294967295>;
opaque verifier_share<0..4294967295>;
case finish:
opaque verifier_message<0..4294967295>;
};
} Message;
/* note that 4294967295 is 2 ** 32 - 1 */
These messages trigger all transitions in the state machine in
Figure 8
, except for the Leader's initial
transition. The Leader's state is initialized using its local inputs with the
following method on class
Vdaf
def ping_pong_leader_init(
self,
vdaf_verify_key: bytes,
ctx: bytes,
agg_param: bytes,
nonce: bytes,
public_share: bytes,
input_share: bytes) -> Continued | Rejected:
"""Called by the Leader to initialize ping-ponging."""
try:
(verify_state, verifier_share) = self.verify_init(
vdaf_verify_key,
ctx,
0,
self.decode_agg_param(agg_param),
nonce,
self.decode_public_share(public_share),
self.decode_input_share(0, input_share),
encoded_verifier_share = self.encode_verifier_share(
verifier_share)
return Continued(
verify_state, 0,
encode(0, encoded_verifier_share), # initialize
except Exception:
return Rejected()
The output is the
State
to which the Leader has transitioned. If the Leader's
state is
Rejected
, then processing halts. Otherwise, if the state is
Continued
, then processing continues. In this case, the state also includes
the Leader's outbound message. The function
encode
is used to encode the
outbound message, which has the message type of
initialize
(identified by the
number
).
To continue processing the report, the Leader sends the outbound message to the
Helper. The Helper's initial transition is computed using the following
procedure:
def ping_pong_helper_init(
self,
vdaf_verify_key: bytes,
ctx: bytes,
agg_param: bytes,
nonce: bytes,
public_share: bytes,
input_share: bytes,
inbound: bytes, # encoded ping pong Message
) -> Continued | FinishedWithOutbound | Rejected:
"""
Called by the Helper in response to the Leader's initial
message.
"""
try:
(verify_state, verifier_share) = self.verify_init(
vdaf_verify_key,
ctx,
1,
self.decode_agg_param(agg_param),
nonce,
self.decode_public_share(public_share),
self.decode_input_share(1, input_share),
(inbound_type, inbound_items) = decode(inbound)
if inbound_type != 0: # initialize
return Rejected()
encoded_verifier_share = inbound_items[0]
verifier_shares = [
self.decode_verifier_share(
verify_state, encoded_verifier_share),
verifier_share,
return self.ping_pong_transition(
ctx, self.decode_agg_param(agg_param),
verifier_shares, verify_state, 0)
except Exception:
return Rejected()
The procedure
decode()
decodes the inbound message and returns the
MessageType variant (
initialize
continue
, or
finish
) and the fields of
the message. The procedure
ping_pong_transition()
takes in the verifier
shares, combines them into the verifier message, and computes the next
verification state of the caller:
def ping_pong_transition(
self,
ctx: bytes,
agg_param: AggParam,
verifier_shares: list[VerifierShare],
verify_state: VerifyState,
verify_round: int) -> Continued | FinishedWithOutbound:
verifier_message = self.verifier_shares_to_message(
ctx, agg_param, verifier_shares)
encoded_verifier_message = self.encode_verifier_message(
verifier_message)
out = self.verify_next(ctx, verify_state, verifier_message)
if verify_round+1 == self.ROUNDS:
return FinishedWithOutbound(
out, encode(2, encoded_verifier_message)) # finalize
(verify_state, verifier_share) = cast(
tuple[VerifyState, VerifierShare], out)
encoded_verifier_share = self.encode_verifier_share(
verifier_share)
return Continued(
verify_state, verify_round+1,
encode(1, encoded_verifier_message,
encoded_verifier_share)) # continue
The output is the
State
to which the Helper has transitioned. If the Helper's
state is
Finished
or
Rejected
, then processing halts. Otherwise, if the
state is
Continued
or
FinishedWithOutbound
, then the state include an
outbound message and processing continues.
To continue processing, the Helper sends the outbound message to the Leader.
The Leader computes its next state transition using the following method on
class
Vdaf
def ping_pong_leader_continued(
self,
ctx: bytes,
agg_param: bytes,
state: Continued,
inbound: bytes, # encoded ping pong Message
) -> State:
"""
Called by the Leader to start the next step of ping-ponging.
"""
return self.ping_pong_continued(
True, ctx, agg_param, state, inbound)
def ping_pong_continued(
self,
is_leader: bool,
ctx: bytes,
agg_param: bytes,
state: Continued,
inbound: bytes, # encoded ping pong Message
) -> State:
try:
verify_round = state.verify_round
(inbound_type, inbound_items) = decode(inbound)
if inbound_type == 0: # initialize
return Rejected()
encoded_verifier_message = inbound_items[0]
verifier_message = self.decode_verifier_message(
state.verify_state,
encoded_verifier_message,
out = self.verify_next(
ctx, state.verify_state, verifier_message)
if verify_round+1 < self.ROUNDS and \
inbound_type == 1: # continue
(verify_state, verifier_share) = cast(
tuple[VerifyState, VerifierShare], out)
encoded_verifier_share = inbound_items[1]
verifier_shares = [
self.decode_verifier_share(
verify_state, encoded_verifier_share),
verifier_share,
if is_leader:
verifier_shares.reverse()
return self.ping_pong_transition(
ctx, self.decode_agg_param(agg_param),
verifier_shares, verify_state, verify_round+1)
elif verify_round+1 == self.ROUNDS and \
inbound_type == 2: # finish
return Finished(out)
else:
return Rejected()
except Exception:
return Rejected()
If the Leader's state is
Finished
or
Rejected
, then processing halts.
Otherwise, if the Leader's state is
Continued
or
FinishedWithOutbound
, the
Leader sends the outbound message to the Helper. The Helper computes its next
state transition using the following method on class
Vdaf
def ping_pong_helper_continued(
self,
ctx: bytes,
agg_param: bytes,
state: Continued,
inbound: bytes, # encoded ping pong Message
) -> State:
"""Called by the Helper to continue ping-ponging."""
return self.ping_pong_continued(
False, ctx, agg_param, state, inbound)
They continue in this way until processing halts. Note that, depending on the
number of rounds of verification that are required, when one party reaches the
Finished
state, there may be one more message to send before the peer can
also finish processing (i.e., the outbound message is not
None
).
5.7.2.
The Star Topology (Any Number of Aggregators)
The ping-pong topology of the previous section is only suitable for applications
of VDAFs involving exactly two Aggregators. In applications with more than two
Aggregators, the star topology described in this section can
be used instead.
Again, one Aggregator initiates the computation. This Aggregator is called the
Leader and all other Aggregators are called Helpers.
At the start of each round, the Leader requests from each Helper its verifier
share. After gathering each of the verifier shares, the Leader computes the
next verifier message (via
vdaf.verifier_shares_to_message()
) and broadcasts
it to the Helpers. At this point, each Aggregator runs
vdaf.verify_next()
locally to either recover an output share or, if more rounds of verification
are required, compute its updated state and verifier share. If another round is
required, then the Helper responds to the broadcast message with its next
verifier share.
The Aggregators proceed in this way until each recovers an output share or some
step of the computation fails.
6.
Preliminaries
This section describes the primitives that are common to the VDAFs specified in
this document.
6.1.
Finite Fields
Both Prio3 and Poplar1 use finite fields of prime order. Finite field
elements are represented by a class
Field
with the following associated
parameters:
MODULUS: int
is the prime modulus that defines the field.
ENCODED_SIZE: int
is the number of bytes used to encode a field element
as a byte string.
Concrete fields, i.e., subclasses of
Field
, implement the following class
methods:
Field.zeros(length: int) -> list[Self]
returns a vector of zeros of the
requested length.
Pre-conditions:
length
MUST
be greater than or equal
Post-conditions:
The length of the output
MUST
be
length
Field.rand_vec(length: int) -> list[Self]
returns a vector of random field
elements and has the same pre- and post-conditions as for
Field.zeros()
Note that this function is not used normatively in the specification of
either Prio3 or Poplar1.
A field element is an instance of a concrete
Field
. Addition,
subtraction, multiplication, division, negation, and inversion are denoted,
respectively,
x + y
x - y
x * y
x / y
-x
, and
x.inv()
Conversion of a field element to an
int
is denoted by
x.int()
. Likewise,
each concrete
Field
implements a constructor for converting an integer into a
field element:
Field(integer: int)
returns
integer
represented as a field element. The
value of
integer
MUST
be in the range
(-Field.MODULUS, Field.MODULUS)
negative values are treated as negations.
6.1.1.
Auxiliary Functions
The following class methods on
Field
are used to encode and decode vectors of
field elements as byte strings:
def encode_vec(cls, vec: list[Self]) -> bytes:
"""
Encode a vector of field elements `vec` as a byte string.
"""
encoded = bytes()
for x in vec:
encoded += to_le_bytes(x.int(), cls.ENCODED_SIZE)
return encoded
def decode_vec(cls, encoded: bytes) -> list[Self]:
"""
Parse a vector of field elements from `encoded`.
"""
if len(encoded) % cls.ENCODED_SIZE != 0:
raise ValueError(
'input length must be a multiple of the size of an '
'encoded field element')
vec = []
while len(encoded) > 0:
(encoded_x, encoded) = front(cls.ENCODED_SIZE, encoded)
x = from_le_bytes(encoded_x)
if x >= cls.MODULUS:
raise ValueError('modulus overflow')
vec.append(cls(x))
return vec
Finally, the following functions define arithmetic on vectors over a finite
field. Note that an exception is raised by each function if the operands are
not the same length.
def vec_sub(left: list[F], right: list[F]) -> list[F]:
"""
Subtract the right operand from the left and return the result.
"""
if len(left) != len(right):
raise ValueError("mismatched vector sizes")
return list(map(lambda x: x[0] - x[1], zip(left, right)))
def vec_add(left: list[F], right: list[F]) -> list[F]:
"""Add the right operand to the left and return the result."""
if len(left) != len(right):
raise ValueError("mismatched vector sizes")
return list(map(lambda x: x[0] + x[1], zip(left, right)))
def vec_neg(vec: list[F]) -> list[F]:
"""Negate the input vector."""
return list(map(lambda x: -x, vec))
6.1.2.
NTT-Friendly Fields
Some VDAFs, including Prio3, require fields that are suitable for efficient
computation of the number theoretic transform (NTT)
Pol71
, as this allows
for fast conversion between polynomial representations. Refer to
SML24
for an overview regarding NTT.
Specifically, a field is said to be "NTT-friendly" if, in addition to the
interface described in
Section 6.1
, it provides the following interface:
Field.gen() -> Self
is a class method that returns the generator of a large
subgroup of the multiplicative group. To be NTT-friendly, the order of this
subgroup
MUST
be a power of two.
GEN_ORDER: int
is the order of the multiplicative subgroup generated by
Field.gen()
. This is the smallest positive integer for which
Field.gen() ** Field.GEN_ORDER == Field(1)
Field.nth_root(n: int) -> F
returns the principal
-th root of
unity that is uniquely determined as
Wn = Field.gen() ** (GEN_ORDER // n)
to ensure interoperability.
Note that
must be a power of two such that
1 <= n <= GEN_ORDER
Field.nth_root_powers(n: int) -> list[F]
returns a list
with
the first
powers of the principal
-th root of unity.
It sets
v[i] = Wn ** i
, such that
Wn = Field.nth_root(n)
Field.ntt(p: list[F], n: int, set_s: bool = False) -> list[F]
returns
a list
with the evaluations of a polynomial
(given its coefficients) at
points.
When
set_s=False
, it sets
v[i] = p(Wn ** i)
(this is the Lagrange
representation of the polynomial, see
Section 6.1.3
);
otherwise, it sets
v[i] = p(s * (Wn ** i))
such that
Wn = Field.nth_root(n)
, and
s = Field.nth_root(2 * n)
Field.inv_ntt(v: list[F], n: int) -> list[F]
returns the coefficients of
a polynomial
such that
are the evaluations of
at the first
powers of the principal
-th root of unity. (This is the monomial
representation of the polynomial, see
Section 6.1.3
.)
The size of the subgroup dictates how large the polynomials can be. It
is
RECOMMENDED
that a generator is chosen with order at least
2**20
6.1.3.
Polynomial Representation
Two standard representations of polynomials are described.
While it is possible to convert between representations
(using the NTT algorithm), polynomials are maintained in the most
efficient representation according to the target operation.
6.1.3.1.
Monomial Basis
The monomial basis is the standard way of representing a polynomial,
where each element of the list corresponds to one of their coefficients
stored in ascending degree order,
that is, starting with the coefficient of the constant term.
The following function denotes polynomial evaluation in the
coefficient representation.
poly_eval(field: type[F], p: list[F], x: F) -> F
returns the
evaluation of the polynomial
(in the monomial basis) at
The
field
parameter is the class object for
and is used by
the implementation to construct field elements. (See
Section 6.1
.)
6.1.3.2.
Lagrange Basis
In the Lagrange basis, polynomials are represented as a list of
evaluations over a set of points.
This set is fixed to the first
powers of the
principal
-th root of unity.
So any polynomial
of degree lesser than
is represented as a
list
of
values such that
v[i] = p(Wn ** i)
where
Wn = Field.nth_root(n)
The
Lagrange
class defines operations over polynomials in the
Lagrange basis. It implements the following methods:
poly_mul(self, p: list[F], q: list[F]) -> list[F]
multiplies two
polynomials, where the input and output polynomials are in the Lagrange basis.
The inputs must have lengths that are equal and a power of two.
poly_eval(self, p: list[F], x: F) -> F
evaluates a polynomial
(in the Lagrange basis) at
in linear time.
It avoids conversion to the monomial basis as shown below.
poly_eval_batched(self, polys: list[list[F]], x: F) -> list[F]
evaluates
every polynomial in the input list at
This function runs faster than invoking
poly_eval
multiple times
because some intermediate calculations are shared.
extend_values_to_power_of_2(self, p: list[F], n: int)
appends evaluations
to the polynomial
(in-place) until the number of evaluations is
The output length
must be a power of two.
double_evaluations(self, p: list[F]) -> list[F]
returns a list of
2*n
evaluations of a polynomial
given
n=len(p)
evaluations.
The length of the input must be a power of two.
An instance of the
Lagrange
class is initialized with an NTT-friendly field.
For algorithm derivations, refer to
Faz25
In the following,
prod
computes the product of its inputs.
class Lagrange[F: NttField]():
def __init__(self, field: type[F]) -> None:
self.field = field
def poly_mul(self, p: list[F], q: list[F]) -> list[F]:
"""Multiply two polynomials in the Lagrange basis."""
n = len(p)
assert_power_of_2(n)
assert len(p) == len(q)
p_2n = self.double_evaluations(p)
q_2n = self.double_evaluations(q)
return [pi*qi for pi, qi in zip(p_2n, q_2n)]
def poly_eval(self, p: list[F], x: F) -> F:
"""Evaluate a polynomial p in the Lagrange basis at x."""
return self.poly_eval_batched([p], x).pop()
def poly_eval_batched(self, polys: list[list[F]], x: F) -> list[F]:
"""Evaluate each polynomial in the Lagrange basis at x."""
assert len({len(p) for p in polys}) == 1
n = len(polys[0])
assert_power_of_2(n)
nodes = self.field.nth_root_powers(n)
k = self.field(1)
u = [p[0] for p in polys]
d = nodes[0] - x
for i in range(1, n):
k *= d
d = nodes[i] - x
t = k * nodes[i]
for j, p in enumerate(polys):
u[j] *= d
if i < len(p):
u[j] += t * p[i]
factor = self.field(-1)**(n-1) * self.field(n).inv()
for i in range(len(u)):
u[i] *= factor
return u
def extend_values_to_power_of_2(self, p: list[F], n: int) -> None:
"""
Appends evaluations to the polynomial p (in-place) until the
number of evaluations is n, and n must be a power of two.
"""
assert_power_of_2(n)
assert len(p) <= n
x = self.field.nth_root_powers(n)
w = [self.field(0)]*n
for i in range(len(p)):
diff = (x[i] - x[j] for j in range(len(p)) if i != j)
w[i] = prod(diff, start=self.field(1))
for k in range(len(p), n):
for i in range(k):
w[i] *= x[i] - x[k]
y_num, y_den = self.field(0), self.field(1)
for i, v in enumerate(p):
y_num = y_num * w[i] + y_den * v
y_den *= w[i]
diff = (x[k] - x[j] for j in range(k))
w[k] = prod(diff, start=self.field(1))
p.append(-w[k] * y_num * y_den.inv())
def double_evaluations(self, p: list[F]) -> list[F]:
"""
Returns 2n evaluations of a polynomial from n Lagrange-basis
evaluations, such that n=len(p) is a power of two.
"""
n = len(p)
assert_power_of_2(n)
even = p
odd = self.field.ntt(self.field.inv_ntt(even, n), n, True)
return [i for pair in zip(even, odd) for i in pair]
6.1.4.
Parameters
Table 4
defines finite fields used in the remainder of this document.
Table 4
Parameters for the finite fields used in this document.
Parameter
Field64
Field128
Field255
MODULUS
2**32 * 4294967295 + 1
2**66 * 4611686018427387897 + 1
2**255 - 19
ENCODED_SIZE
16
32
Generator
7**4294967295
7**4611686018427387897
n/a
GEN_ORDER
2**32
2**66
n/a
6.2.
Extendable Output Functions (XOFs)
VDAFs in this specification use eXtendable Output Functions (XOFs) for two
purposes:
Extracting short, pseudorandom strings called "seeds" from high entropy
inputs
Expanding seeds into long, pseudorandom outputs
Concrete XOFs implement a class
Xof
providing the following interface:
SEED_SIZE: int
is the size (in bytes) of a seed.
Xof(seed: bytes, dst: bytes, binder: bytes)
constructs an instance of the
XOF from the given seed and a domain separation tag and binder string as
defined in
Section 6.2.3
. The length of the seed will typically be
SEED_SIZE
, but some XOFs may support multiple seed sizes. The seed
MUST
be
generated securely, i.e., it is either the output of a CSPRNG or a
previous invocation of the XOF.
xof.next(length: int)
returns the next chunk of the output of the
initialized XOF as a byte string. The length of the chunk
MUST
be
length
The following methods are provided for all concrete XOFs. The first is a class
method used to derive a fresh seed from an existing one. The second is an
instance method used to compute a sequence of field elements. The third is a
class method that provides a one-shot interface for expanding a seed into a
field vector.
def derive_seed(cls,
seed: bytes,
dst: bytes,
binder: bytes) -> bytes:
"""
Derive a new seed.
Pre-conditions:
- `len(seed) == cls.SEED_SIZE`
"""
xof = cls(seed, dst, binder)
return xof.next(cls.SEED_SIZE)
def next_vec(self, field: type[F], length: int) -> list[F]:
"""
Output the next `length` field elements.
Pre-conditions:
- `field` is sub-class of `Field`
- `length > 0`
"""
m = next_power_of_2(field.MODULUS) - 1
vec: list[F] = []
while len(vec) < length:
x = from_le_bytes(self.next(field.ENCODED_SIZE))
x &= m
if x < field.MODULUS:
vec.append(field(x))
return vec
def expand_into_vec(cls,
field: type[F],
seed: bytes,
dst: bytes,
binder: bytes,
length: int) -> list[F]:
"""
Expand the input `seed` into a vector of `length` field elements.
Pre-conditions:
- `field` is sub-class of `Field`
- `len(seed) == cls.SEED_SIZE`
- `length > 0`
"""
xof = cls(seed, dst, binder)
return xof.next_vec(field, length)
6.2.1.
XofTurboShake128
This section describes XofTurboShake128, an XOF based on the TurboSHAKE128
function specified in
RFC9861
. This XOF is
RECOMMENDED
for all use cases
for DAFs and VDAFs.
Pre-conditions:
The default seed length is
32
. The seed
MAY
have a different length, but it
MUST
not exceed 255. Otherwise initialization will raise an exception.
The length of the domain separation string
dst
passed to XofTurboShake128
MUST NOT
exceed 65535 bytes. Otherwise initialization will raise an
exception.
class XofTurboShake128(Xof):
"""XOF wrapper for TurboSHAKE128."""
# Associated parameters
SEED_SIZE = 32
def __init__(self, seed: bytes, dst: bytes, binder: bytes):
self.l = 0
self.m = \
to_le_bytes(len(dst), 2) + dst \
to_le_bytes(len(seed), 1) + seed + \
binder
def next(self, length: int) -> bytes:
self.l += length
# Function `TurboSHAKE128(M, D, L)` is as defined in
# Section 2.1 of [RFC9861].
# Implementation note: rather than re-generate the output
# stream each time `next()` is invoked, most implementations
# of TurboSHAKE128 will expose an "absorb-then-squeeze" API
# that allows stateful handling of the stream.
stream = TurboSHAKE128(self.m, 1, self.l)
return stream[-length:]
6.2.2.
XofFixedKeyAes128
The XOF in the previous section can be used safely wherever a XOF is needed in
this document. However, there are some situations where TurboSHAKE128 creates a
performance bottleneck and a more efficient XOF can be used safely instead.
This section describes XofFixedKeyAes128, which is used to implement the IDPF
of Poplar1 (
Section 8.3
). It is
NOT RECOMMENDED
to use this XOF for any
other purpose. See
Section 9.6
for a more detailed discussion.
XofFixedKeyAes128 uses the AES-128 blockcipher
AES
for most of the
computation, thereby taking advantage of the hardware implementations of this
blockcipher that are widely available. AES-128 is used in a fixed-key mode of
operation; the key is derived during initialization using TurboSHAKE128.
Pre-conditions:
The length of the seed
MUST
be
16
The length of the domain separation string
dst
passed to XofFixedKeyAes128
MUST NOT
exceed 65535 bytes. Otherwise initialization will raise an
exception.
class XofFixedKeyAes128(Xof):
"""
XOF based on a circular collision-resistant hash function from
fixed-key AES.
"""
# Associated parameters
SEED_SIZE = 16
def __init__(self, seed: bytes, dst: bytes, binder: bytes):
if len(seed) != self.SEED_SIZE:
raise ValueError("incorrect seed size")
self.length_consumed = 0
# Use TurboSHAKE128 to derive a key from the binder string
# and domain separation tag. Note that the AES key does not
# need to be kept secret from any party. However, when used
# with an IDPF, we require the binder to be a random nonce.
# Implementation note: this step can be cached across XOF
# evaluations with many different seeds.
dst_length = to_le_bytes(len(dst), 2)
self.fixed_key = TurboSHAKE128(
dst_length + dst + binder,
2,
16,
self.seed = seed
def next(self, length: int) -> bytes:
offset = self.length_consumed % 16
new_length = self.length_consumed + length
block_range = range(
self.length_consumed // 16,
new_length // 16 + 1
self.length_consumed = new_length
hashed_blocks = [
self.hash_block(xor(self.seed, to_le_bytes(i, 16)))
for i in block_range
return concat(hashed_blocks)[offset:offset+length]
def hash_block(self, block: bytes) -> bytes:
"""
The multi-instance tweakable circular correlation-robust hash
function of [GKWWY20] (Section 4.2). The tweak here is the
key that stays constant for all XOF evaluations of the same
Client, but differs between Clients.
Function `AES128(key, block)` is the AES-128 blockcipher.
"""
lo, hi = block[:8], block[8:]
sigma_block = concat([hi, xor(hi, lo)])
return xor(AES128(self.fixed_key, sigma_block), sigma_block)
6.2.3.
The Domain Separation Tag and Binder String
XOFs are used to map a seed to a finite domain, e.g., a fresh seed or a vector
of field elements. To ensure domain separation, derivation is bound to some
distinguished domain separation tag. The domain separation tag encodes the
following values:
The document version (i.e.,
VERSION
);
The "class" of the algorithm using the output (e.g., DAF, VDAF, or IDPF as
defined in
Section 8.1
);
A unique identifier for the algorithm (e.g.,
VDAF.ID
); and
Some indication of how the output is used (e.g., for deriving the measurement
shares in Prio3
Section 7
).
The following algorithm is used in the remainder of this document in order to
format the domain separation tag:
def format_dst(algo_class: int,
algo: int,
usage: int) -> bytes:
"""
Format XOF domain separation tag.
Pre-conditions:
- `algo_class` in the range `[0, 2**8)`
- `algo` in the range `[0, 2**32)`
- `usage` in the range `[0, 2**16)`
"""
return concat([
to_be_bytes(VERSION, 1),
to_be_bytes(algo_class, 1),
to_be_bytes(algo, 4),
to_be_bytes(usage, 2),
])
It is also sometimes necessary to bind the output to some ephemeral value that
multiple parties need to agree on. This input is called the "binder string".
7.
Prio3
This section describes Prio3, a VDAF for general-purpose aggregation. Prio3 is
suitable for a wide variety of aggregation functions, including (but not
limited to) sum, mean, standard deviation, histograms, and linear regression.
It is compatible with any aggregation function that has the following
structure:
Each measurement is encoded as a vector over some finite field.
Measurement validity is determined by an "arithmetic circuit" evaluated over
the encoded measurement. An arithmetic circuit is a function comprised of
arithmetic operations in the field. (These are specified in full detail in
Section 7.3.2
.)
The aggregate result is obtained by summing up the encoded measurements and
computing some function of the sum.
Clients protect the privacy of their measurements by secret sharing them and
distributing the shares among the Aggregators. To ensure each measurement is
valid, the Aggregators run a multi-party computation on their shares, the
result of which is the output of the arithmetic circuit. This involves
verification of a "Fully Linear Proof (FLP)" (
Section 7.1
) generated by the Client.
FLPs are the core component of Prio3, as they specify the types of
measurements and how they are encoded, verified, and aggregated. In fact Prio3
can be thought of as a transformation of an FLP into a VDAF.
Prio3 does not have an aggregation parameter. Instead, each output share is
derived from each input share by applying a fixed map. See
Section 8
for an
example of a VDAF that makes meaningful use of the aggregation parameter.
The remainder of this section is structured as follows. The interface of FLPs is
described in
Section 7.1
. The generic transformation of an FLP into Prio3 is
specified in
Section 7.2
. Next, a concrete FLP suitable for any
validity circuit is specified in
Section 7.3
. Finally, variants of Prio3
for various types of aggregation tasks are specified in
Section 7.4
. Test vectors for each variant can be found in
Appendix C
7.1.
Fully Linear Proofs (FLPs)
Conceptually, an FLP is a two-party protocol executed by a prover and a
verifier. The verifier is restricted to only access the messages it receives
from the prover via linear queries. In actual use in Prio3, however, the
prover's computation is carried out by the Client, and the verifier's
computation is distributed among the Aggregators. The Client generates a
"proof" of its measurement's validity and distributes shares of the proof to
the Aggregators. During verification, each Aggregator performs some computation
on its measurement share and proof share locally, then broadcasts the result in
its verifier share. The validity decision is then made by the
verifier_shares_to_message()
algorithm (
Section 5.2
).
As usual, the interface implemented by a concrete FLP is described in terms of
an object
flp
of type
Flp
that specifies the set of methods and parameters
a concrete FLP must provide.
The parameters provided by a concrete FLP are listed in
Table 5
. A
concrete FLP specifies the following algorithms for generating and verifying
proofs of validity (encoding is described below in
Section 7.1.1
):
flp.prove(meas: list[F], prove_rand: list[F], joint_rand: list[F]) ->
list[F]
is the proof-generation algorithm run by the prover. Its inputs are
the encoded measurement, the "prover randomness"
prove_rand
, and the "joint
randomness"
joint_rand
. The prover randomness is used only by the prover,
but the joint randomness is shared by both the prover and verifier.
flp.query(meas: list[F], proof: list[F], query_rand: list[F], joint_rand:
list[F], num_shares: int) -> list[F]
is the linear query algorithm run by the
verifier on the encoded measurement and proof. The result of the query (i.e.,
the output of this function) is called the "verifier message". In addition to
the measurement and proof, this algorithm takes as input the query randomness
query_rand
and the joint randomness
joint_rand
. The former is used only
by the verifier.
num_shares
specifies the number of shares (more on this
below).
flp.decide(verifier: list[F]) -> bool
is the deterministic decision
algorithm run by the verifier. It takes as input the verifier message and
outputs a boolean indicating if the measurement from which it was generated
is valid.
This application requires that the FLP is "fully linear" in the sense defined in
BBCGGI19
. As a practical matter, what this property implies is that, when
run on a share of the measurement and proof, the query algorithm outputs a
share of the verifier message (hereafter the "verifier share"). Furthermore,
the privacy property of the FLP system ensures that the verifier message
reveals nothing about the measurement other than the fact that it is valid.
Therefore, to decide if a measurement is valid, the Aggregators will run the
query algorithm locally, exchange verifier shares, combine them to recover the
verifier message, and run the decision algorithm.
The query algorithm includes a parameter
num_shares
that specifies the number
of shares of the measurement and proof that were generated. If these data are
not secret shared, then
num_shares == 1
. This parameter is useful for
normalizing constants in arithmetic circuits so that each Aggregator properly
computes a secret share of the circuit's output. See
Section 7.3
for
details.
An FLP is executed by the prover and verifier as follows:
def run_flp(
flp: Flp[Measurement, AggResult, F],
meas: list[F],
num_shares: int) -> bool:
"""Run the FLP on an encoded measurement."""
joint_rand = flp.field.rand_vec(flp.JOINT_RAND_LEN)
prove_rand = flp.field.rand_vec(flp.PROVE_RAND_LEN)
query_rand = flp.field.rand_vec(flp.QUERY_RAND_LEN)
# Prover generates the proof.
proof = flp.prove(meas, prove_rand, joint_rand)
# Shard the measurement and the proof.
meas_shares = additive_secret_share(
meas,
num_shares,
flp.field,
proof_shares = additive_secret_share(
proof,
num_shares,
flp.field,
# Verifier queries the meas shares and proof shares.
verifier_shares = [
flp.query(
meas_share,
proof_share,
query_rand,
joint_rand,
num_shares,
for meas_share, proof_share in zip(meas_shares, proof_shares)
# Combine the verifier shares into the verifier.
verifier = flp.field.zeros(len(verifier_shares[0]))
for verifier_share in verifier_shares:
verifier = vec_add(verifier, verifier_share)
# Verifier decides if the measurement is valid.
return flp.decide(verifier)
The proof system is designed so that, if
meas
is valid, then
run_flp(flp,
meas, num_shares)
always returns
True
. On the other hand, if
meas
is
invalid, then as long as
joint_rand
and
query_rand
are generated uniform
randomly, the output is
False
with high probability. False positives are
possible: there is a small probability that a verifier accepts an invalid input
as valid. An FLP is said to be "sound" if this probability is sufficiently
small. The soundness of the FLP depends on a variety of parameters, like the
length of the input and the size of the field. See
Section 7.3
for
details.
Note that soundness of an FLP system is not the same as verifiability for the
VDAF that uses it. In particular, soundness of the FLP is necessary, but
insufficient for verifiability of Prio3 (
Section 7
). See
Section 9.7
for details.
In addition, note that
BBCGGI19
defines a larger class of fully linear
proof systems than is considered here. In particular, what is called an "FLP"
here is called a 1.5-round, public-coin, interactive oracle proof system in
their paper.
Table 5
FLP parameters.
Parameter
Description
PROVE_RAND_LEN: int
Length of the prover randomness, the number of random field elements consumed by the prover when generating a proof.
QUERY_RAND_LEN: int
Length of the query randomness, the number of random field elements consumed by the verifier.
JOINT_RAND_LEN: int
Length of the joint randomness, the number of random field elements shared by the prover and verifier.
MEAS_LEN: int
Length of the encoded measurement (
Section 7.1.1
).
OUTPUT_LEN: int
Length of the aggregatable output (
Section 7.1.1
).
PROOF_LEN: int
Length of the proof.
VERIFIER_LEN: int
Length of the verifier message generated by querying the measurement and proof.
Measurement
Type of the measurement.
AggResult
Type of the aggregate result.
field: type[F]
Class object for the field (
Section 6.1
).
7.1.1.
Encoding the Input
The type of measurement being aggregated is defined by the FLP. Hence, the FLP
also specifies a method of encoding raw measurements as a vector of field
elements:
flp.encode(measurement: Measurement) -> list[F]
encodes a raw measurement
as a vector of field elements.
Post-conditions:
The encoded measurement
MUST
have length
flp.MEAS_LEN
For some FLPs, the encoded measurement also includes redundant field elements
that are useful for checking the proof, but which are not needed after the
proof has been checked. An example is the
Sum
type defined in
Section 7.4.2
for which each measurement is an integer in the range
[0, max_measurement]
The range check requires encoding the measurement with several field elements,
though just one is needed for aggregation. Thus the FLP defines an algorithm
for truncating the encoded measurement to the length of the aggregatable output:
flp.truncate(meas: list[F]) -> list[F]
maps an encoded measurement (e.g.,
the bit-encoding of the measurement) to an aggregatable output (e.g., the
singleton vector containing the measurement).
Pre-conditions:
The length of the input
MUST
be
flp.MEAS_LEN
Post-conditions:
The length of the output
MUST
be
flp.OUTPUT_LEN
Once the aggregate shares have been transmitted to the Collector, their sum can
be converted into the aggregate result. This could be a projection from the
FLP's field to the integers, or it could include additional post-processing.
Either way, this functionality is implemented by the following method:
flp.decode(output: list[F], num_measurements: int) -> AggResult
maps a sum
of aggregate shares to an aggregate result.
Pre-conditions:
The length of the output
MUST
be
OUTPUT_LEN
num_measurements
MUST
equal the number of measurements that were
aggregated.
Taken together, these three functionalities correspond to the notion of
"Affine-aggregatable encodings (AFEs)" from
CGB17
7.1.2.
Multiple Proofs
It is sometimes desirable to generate and verify multiple independent proofs
for the same input. First, this improves the soundness of the proof system
without having to change any of its parameters. Second, it allows a smaller
field to be used (e.g., replace Field128 with Field64)
without sacrificing soundness. This is useful because it reduces the overall
communication of the protocol. (This is a trade-off, of course, since
generating and verifying more proofs requires more time.) Given these benefits,
this feature is implemented by Prio3 (
Section 7
).
To generate these proofs for a specific measurement, the prover calls
flp.prove()
multiple times, each time using fresh prover and joint
randomness. The verifier checks each proof independently, each time with fresh
query randomness. It accepts the measurement only if the decision algorithm
accepts on each proof.
See
Section 9.7
for guidance on choosing the field size and number
of proofs.
7.2.
Specification
This section specifies
Prio3
, an implementation of the
Vdaf
interface
defined in
Section 5
. The parameters and types required by the
Vdaf
interface
are defined in
Table 6
. The methods required for sharding,
verification, aggregation, and unsharding are described in the remaining
subsections. These methods refer to constants enumerated in
Table 7
Table 6
Parameters for Prio3.
Parameter
Value
flp
An instance of
Flp
Section 7.1
).
xof
XofTurboShake128
Section 6.2.1
PROOFS
Any
int
in the range
[1, 256)
VERIFY_KEY_SIZE
xof.SEED_SIZE
RAND_SIZE
xof.SEED_SIZE * SHARES if flp.JOINT_RAND_LEN == 0 else 2 * xof.SEED_SIZE * SHARES
NONCE_SIZE
16
ROUNDS
SHARES
Any
int
in the range
[2, 256)
Measurement
As defined by
flp
AggParam
None
PublicShare
Optional[list[bytes]]
InputShare
tuple[list[F], list[F], Optional[bytes]] | tuple[bytes, Optional[bytes]]
OutShare
list[F]
AggShare
list[F]
AggResult
As defined by
flp
VerifyState
tuple[list[F], Optional[bytes]]
VerifierShare
tuple[list[F], Optional[bytes]]
VerifierMessage
Optional[bytes]
Table 7
Constants used by Prio3.
Variable
Value
USAGE_MEAS_SHARE: int
USAGE_PROOF_SHARE: int
USAGE_JOINT_RANDOMNESS: int
USAGE_PROVE_RANDOMNESS: int
USAGE_QUERY_RANDOMNESS: int
USAGE_JOINT_RAND_SEED: int
USAGE_JOINT_RAND_PART: int
7.2.1.
Sharding
Recall from
Section 7.1
that the FLP syntax calls for "joint randomness" shared by
the prover (i.e., the Client) and the verifier (i.e., the Aggregators). VDAFs
have no such notion. Instead, the Client derives the joint randomness from its
measurement in a way that allows the Aggregators to reconstruct it from their
shares. (This idea is based on the Fiat-Shamir heuristic and is described in
Section 6.2.3 of
BBCGGI19
.)
The sharding algorithm involves the following steps:
Encode the Client's measurement as specified by the FLP
Shard the measurement into a sequence of measurement shares
Derive the joint randomness from the measurement shares and nonce
Generate the proof using the derived joint randomness
Shard the proof into a sequence of proof shares
As described in
Section 7.1.2
, the probability of an invalid measurement
being deemed valid can be decreased by generating and verifying multiple
proofs. To support this:
In step 3, derive as much joint randomness as required by
PROOFS
proofs
Repeat step 4
PROOFS
times, each time with a unique joint randomness
Depending on the FLP, joint randomness may not be required. In particular, when
flp.JOINT_RAND_LEN == 0
, the Client does not derive the joint randomness
(Step 3).
The sharding algorithm is specified below:
def shard(
self,
ctx: bytes,
measurement: Measurement,
nonce: bytes,
rand: bytes) -> tuple[
Optional[list[bytes]],
list[Prio3InputShare]]:
if len(nonce) != self.NONCE_SIZE:
raise ValueError("incorrect nonce size")
if len(rand) != self.RAND_SIZE:
raise ValueError("incorrect size of random bytes argument")
l = self.xof.SEED_SIZE
seeds = [rand[i:i + l] for i in range(0, self.RAND_SIZE, l)]
meas = self.flp.encode(measurement)
if self.flp.JOINT_RAND_LEN > 0:
return self.shard_with_joint_rand(ctx, meas, nonce, seeds)
else:
return self.shard_without_joint_rand(ctx, meas, seeds)
It starts by splitting the randomness into seeds. It then encodes the
measurement as prescribed by the FLP and calls one of two methods, depending on
whether joint randomness is required by the FLP. The methods are defined in the
subsections below.
7.2.1.1.
FLPs Without Joint Randomness
The following method is used for FLPs that do not require joint randomness,
i.e., when
flp.JOINT_RAND_LEN == 0
. It consists of the following steps:
Shard the encoded measurement into shares
Generate proofs and shard each into shares
Encode each measurement share and shares of each proof into an input share
Only one pair of measurement and proof(s) share (called the "Leader" shares)
are vectors of field elements. The other shares (called the "Helper"
shares) are represented instead by an XOF seed, which is expanded into vectors
of field elements. The methods on
Prio3
for deriving the prover randomness,
measurement shares, and proof shares are defined in
Section 7.2.6
def shard_without_joint_rand(
self,
ctx: bytes,
meas: list[F],
seeds: list[bytes]) -> tuple[
Optional[list[bytes]],
list[Prio3InputShare[F]]]:
helper_shares, seeds = front(self.SHARES - 1, seeds)
(prove_seed,), seeds = front(1, seeds)
# Shard the encoded measurement into shares.
leader_meas_share = meas
for j in range(self.SHARES - 1):
leader_meas_share = vec_sub(
leader_meas_share,
self.helper_meas_share(ctx, j + 1, helper_shares[j]),
# Generate and shard each proof into shares.
prove_rands = self.prove_rands(ctx, prove_seed)
leader_proofs_share = []
for _ in range(self.PROOFS):
prove_rand, prove_rands = front(
self.flp.PROVE_RAND_LEN, prove_rands)
leader_proofs_share += self.flp.prove(meas, prove_rand, [])
for j in range(self.SHARES - 1):
leader_proofs_share = vec_sub(
leader_proofs_share,
self.helper_proofs_share(
ctx,
j + 1,
helper_shares[j],
),
# Each Aggregator's input share contains its measurement share
# and its share of the proof(s).
input_shares: list[Prio3InputShare[F]] = []
input_shares.append((
leader_meas_share,
leader_proofs_share,
None,
))
for j in range(self.SHARES - 1):
input_shares.append((
helper_shares[j],
None,
))
return (None, input_shares)
7.2.1.2.
FLPs With Joint Randomness
The following method is used for FLPs that require joint randomness, i.e., for
which
flp.JOINT_RAND_LEN > 0
. Joint randomness derivation involves an
additional XOF seed for each Aggregator called the "blind". The computation
involves the following steps:
Compute a "joint randomness part" from each measurement share and blind
Compute a "joint randomness seed" from the joint randomness parts
Compute the joint randomness for each proof evaluation from the joint
randomness seed
This three-step process is designed to ensure that the joint randomness does
not leak the measurement to the Aggregators while preventing a malicious Client
from tampering with the joint randomness in a way that causes the Aggregators
to accept an invalid measurement. To save a round of communication between the
Aggregators later, the Client encodes the joint randomness parts in the public
share. (See
Section 7.2.2
for details.)
All functions used in the following listing are defined in
Section 7.2.6
def shard_with_joint_rand(
self,
ctx: bytes,
meas: list[F],
nonce: bytes,
seeds: list[bytes]) -> tuple[
Optional[list[bytes]],
list[Prio3InputShare[F]]]:
helper_seeds, seeds = front((self.SHARES - 1) * 2, seeds)
helper_shares = [
helper_seeds[i]
for i in range(0, (self.SHARES - 1) * 2, 2)
helper_blinds = [
helper_seeds[i]
for i in range(1, (self.SHARES - 1) * 2, 2)
(leader_blind, prove_seed), seeds = front(2, seeds)
# Shard the encoded measurement into shares and compute the
# joint randomness parts.
leader_meas_share = meas
joint_rand_parts = []
for j in range(self.SHARES - 1):
helper_meas_share = self.helper_meas_share(
ctx, j + 1, helper_shares[j])
leader_meas_share = vec_sub(leader_meas_share,
helper_meas_share)
joint_rand_parts.append(self.joint_rand_part(
ctx, j + 1, helper_blinds[j],
helper_meas_share, nonce))
joint_rand_parts.insert(0, self.joint_rand_part(
ctx, 0, leader_blind, leader_meas_share, nonce))
# Generate each proof and shard it into proof shares.
prove_rands = self.prove_rands(ctx, prove_seed)
joint_rands = self.joint_rands(
ctx, self.joint_rand_seed(ctx, joint_rand_parts))
leader_proofs_share = []
for _ in range(self.PROOFS):
prove_rand, prove_rands = front(
self.flp.PROVE_RAND_LEN, prove_rands)
joint_rand, joint_rands = front(
self.flp.JOINT_RAND_LEN, joint_rands)
leader_proofs_share += self.flp.prove(
meas,
prove_rand,
joint_rand,
for j in range(self.SHARES - 1):
leader_proofs_share = vec_sub(
leader_proofs_share,
self.helper_proofs_share(
ctx,
j + 1,
helper_shares[j],
),
# Each Aggregator's input share contains its measurement share,
# share of proof(s), and blind. The public share contains the
# Aggregators' joint randomness parts.
input_shares: list[Prio3InputShare[F]] = []
input_shares.append((
leader_meas_share,
leader_proofs_share,
leader_blind,
))
for j in range(self.SHARES - 1):
input_shares.append((
helper_shares[j],
helper_blinds[j],
))
return (joint_rand_parts, input_shares)
7.2.2.
Verification
This section describes the process of recovering output shares from the input
shares. The high-level idea is that each Aggregator first queries its
measurement share and proof(s) share(s) locally, then broadcasts its verifier
share(s). The shares of verifier(s) are then combined into the verifier
message(s) used to decide whether to accept.
In addition, the Aggregators must recompute the same joint randomness used by
the Client to generate the proof(s). In order to avoid an extra round of
communication, the Client includes the joint randomness parts in the public
share. This leaves open the possibility that the Client cheated by, say,
forcing the Aggregators to use joint randomness that biases the proof check
procedure some way in its favor. To mitigate this, the Aggregators also check
that they have all computed the same joint randomness seed before accepting
their output shares. To do so, they exchange their parts of the joint
randomness along with their shares of verifier(s).
Implementation note: the verification state for Prio3 includes the output share
that will be released once verification is complete. In some situations, it may
be necessary for the Aggregator to encode this state as bytes and store it for
retrieval later on. For all but the first Aggregator, it is possible to save
storage by storing the measurement share rather than output share itself. It is
relatively inexpensive to expand this seed into the measurement share, then
truncate the measurement share to get the output share.
All functions used in the following listing are defined in
Section 7.2.6
def verify_init(
self,
verify_key: bytes,
ctx: bytes,
agg_id: int,
_agg_param: None,
nonce: bytes,
public_share: Optional[list[bytes]],
input_share: Prio3InputShare[F]) -> tuple[
Prio3VerifyState[F],
Prio3VerifierShare[F]]:
joint_rand_parts = public_share
(meas_share, proofs_share, blind) = \
self.expand_input_share(ctx, agg_id, input_share)
out_share = self.flp.truncate(meas_share)
# Compute the joint randomness.
joint_rand: list[F] = []
corrected_joint_rand_seed, joint_rand_part = None, None
if self.flp.JOINT_RAND_LEN > 0:
assert blind is not None
assert joint_rand_parts is not None
joint_rand_part = self.joint_rand_part(
ctx, agg_id, blind, meas_share, nonce)
joint_rand_parts = list(joint_rand_parts)
joint_rand_parts[agg_id] = joint_rand_part
corrected_joint_rand_seed = self.joint_rand_seed(
ctx, joint_rand_parts)
joint_rands = self.joint_rands(
ctx, corrected_joint_rand_seed)
# Query the measurement and proof(s) share(s).
query_rands = self.query_rands(verify_key, ctx, nonce)
verifiers_share = []
for _ in range(self.PROOFS):
proof_share, proofs_share = front(
self.flp.PROOF_LEN, proofs_share)
query_rand, query_rands = front(
self.flp.QUERY_RAND_LEN, query_rands)
if self.flp.JOINT_RAND_LEN > 0:
joint_rand, joint_rands = front(
self.flp.JOINT_RAND_LEN, joint_rands)
verifiers_share += self.flp.query(
meas_share,
proof_share,
query_rand,
joint_rand,
self.SHARES,
verify_state = (out_share, corrected_joint_rand_seed)
verifier_share = (verifiers_share, joint_rand_part)
return (verify_state, verifier_share)
def verifier_shares_to_message(
self,
ctx: bytes,
_agg_param: None,
verifier_shares: list[Prio3VerifierShare[F]],
) -> Optional[bytes]:
# Unshard each set of verifier shares into each verifier message.
verifiers = self.flp.field.zeros(
self.flp.VERIFIER_LEN * self.PROOFS)
joint_rand_parts = []
for (verifiers_share, joint_rand_part) in verifier_shares:
verifiers = vec_add(verifiers, verifiers_share)
if self.flp.JOINT_RAND_LEN > 0:
assert joint_rand_part is not None
joint_rand_parts.append(joint_rand_part)
# Verify that each proof is well-formed and input is valid.
for _ in range(self.PROOFS):
verifier, verifiers = front(self.flp.VERIFIER_LEN, verifiers)
if not self.flp.decide(verifier):
raise ValueError('proof verifier check failed')
# Combine the joint randomness parts computed by the
# Aggregators into the true joint randomness seed. This is
# used in the last step.
joint_rand_seed = None
if self.flp.JOINT_RAND_LEN > 0:
joint_rand_seed = self.joint_rand_seed(ctx, joint_rand_parts)
return joint_rand_seed
def verify_next(
self,
_ctx: bytes,
verify_state: Prio3VerifyState[F],
verifier_message: Optional[bytes]
) -> tuple[Prio3VerifyState[F], Prio3VerifierShare[F]] | list[F]:
joint_rand_seed = verifier_message
(out_share, corrected_joint_rand_seed) = verify_state
# If joint randomness was used, check that the value computed by
# the Aggregators matches the value indicated by the Client.
if joint_rand_seed != corrected_joint_rand_seed:
raise ValueError('joint randomness check failed')
return out_share
7.2.3.
Validity of Aggregation Parameters
Prio3
only permits a report to be aggregated once.
def is_valid(
self,
_agg_param: None,
previous_agg_params: list[None]) -> bool:
return len(previous_agg_params) == 0
7.2.4.
Aggregation
Aggregating a set of output shares is simply a matter of adding up the vectors
element-wise.
def agg_init(self, _agg_param: None) -> list[F]:
return self.flp.field.zeros(self.flp.OUTPUT_LEN)
def agg_update(self,
_agg_param: None,
agg_share: list[F],
out_share: list[F]) -> list[F]:
return vec_add(agg_share, out_share)
def merge(self,
_agg_param: None,
agg_shares: list[list[F]]) -> list[F]:
agg = self.agg_init(None)
for agg_share in agg_shares:
agg = vec_add(agg, agg_share)
return agg
7.2.5.
Unsharding
To unshard a set of aggregate shares, the Collector first adds up the vectors
element-wise, then decodes the aggregate result from the sum according to the
FLP (
Section 7.1.1
).
def unshard(
self,
_agg_param: None,
agg_shares: list[list[F]],
num_measurements: int) -> AggResult:
agg = self.merge(None, agg_shares)
return self.flp.decode(agg, num_measurements)
7.2.6.
Auxiliary Functions
This section defines a number of auxiliary functions referenced by the main
algorithms for Prio3 in the preceding sections.
def helper_meas_share(
self,
ctx: bytes,
agg_id: int,
share: bytes) -> list[F]:
return self.xof.expand_into_vec(
self.flp.field,
share,
self.domain_separation_tag(USAGE_MEAS_SHARE, ctx),
byte(agg_id),
self.flp.MEAS_LEN,
def helper_proofs_share(
self,
ctx: bytes,
agg_id: int,
share: bytes) -> list[F]:
return self.xof.expand_into_vec(
self.flp.field,
share,
self.domain_separation_tag(USAGE_PROOF_SHARE, ctx),
byte(self.PROOFS) + byte(agg_id),
self.flp.PROOF_LEN * self.PROOFS,
def expand_input_share(
self,
ctx: bytes,
agg_id: int,
input_share: Prio3InputShare[F]) -> tuple[
list[F],
list[F],
Optional[bytes]]:
if agg_id > 0:
assert len(input_share) == 2
(share, blind) = input_share
meas_share = self.helper_meas_share(ctx, agg_id, share)
proofs_share = self.helper_proofs_share(ctx, agg_id, share)
else:
assert len(input_share) == 3
(meas_share, proofs_share, blind) = input_share
return (meas_share, proofs_share, blind)
def prove_rands(self, ctx: bytes, prove_seed: bytes) -> list[F]:
return self.xof.expand_into_vec(
self.flp.field,
prove_seed,
self.domain_separation_tag(USAGE_PROVE_RANDOMNESS, ctx),
byte(self.PROOFS),
self.flp.PROVE_RAND_LEN * self.PROOFS,
def query_rands(
self,
verify_key: bytes,
ctx: bytes,
nonce: bytes) -> list[F]:
return self.xof.expand_into_vec(
self.flp.field,
verify_key,
self.domain_separation_tag(USAGE_QUERY_RANDOMNESS, ctx),
byte(self.PROOFS) + nonce,
self.flp.QUERY_RAND_LEN * self.PROOFS,
def joint_rand_part(
self,
ctx: bytes,
agg_id: int,
blind: bytes,
meas_share: list[F],
nonce: bytes) -> bytes:
return self.xof.derive_seed(
blind,
self.domain_separation_tag(USAGE_JOINT_RAND_PART, ctx),
byte(agg_id) + nonce + self.flp.field.encode_vec(meas_share),
def joint_rand_seed(self,
ctx: bytes,
joint_rand_parts: list[bytes]) -> bytes:
"""Derive the joint randomness seed from its parts."""
return self.xof.derive_seed(
zeros(self.xof.SEED_SIZE),
self.domain_separation_tag(USAGE_JOINT_RAND_SEED, ctx),
concat(joint_rand_parts),
def joint_rands(self,
ctx: bytes,
joint_rand_seed: bytes) -> list[F]:
"""Derive the joint randomness from its seed."""
return self.xof.expand_into_vec(
self.flp.field,
joint_rand_seed,
self.domain_separation_tag(USAGE_JOINT_RANDOMNESS, ctx),
byte(self.PROOFS),
self.flp.JOINT_RAND_LEN * self.PROOFS,
7.2.7.
Message Serialization
This section defines serialization formats for messages exchanged over the
network while executing Prio3. Messages are defined in the presentation
language of TLS as defined in
Section 3
of [
RFC8446
Let
prio3
denote an instance of
Prio3
. In the remainder, let
be an
alias for
prio3.xof.SEED_SIZE
and
as an alias for
prio3.field.ENCODED_SIZE
. XOF seeds are represented as follows:
opaque Prio3Seed[S];
Field elements are encoded in little-endian byte order (as defined in
Section 6.1
) and represented as follows:
opaque Prio3Field[F];
7.2.7.1.
Public Share
The contents of the public share depend on whether joint randomness is
required for the underlying FLP (i.e.,
prio3.flp.JOINT_RAND_LEN > 0
). If
joint randomness is not used, then the public share is the empty string.
Otherwise, if joint randomness is used, then the public share encodes the joint
randomness parts as follows:
struct {
Prio3Seed joint_rand_parts[S * prio3.SHARES];
} Prio3PublicShareWithJointRand;
7.2.7.2.
Input Share
Just as for the public share, the content of the input shares depends on
whether joint randomness is used. If so, then each input share includes the
Aggregator's blind for generating its joint randomness part.
In addition, the encoding of the input shares depends on which aggregator is
receiving the message. If the aggregator ID is
, then the input share
includes the full measurement share and proofs(s) share(s). Otherwise, if the
aggregator ID is greater than
, then the measurement and shares of proof(s)
are represented by an XOF seed. Just as in
Section 5.7.2
, the former is called
the Leader and the latter the Helpers.
In total there are four variants of the input share. When joint randomness is
not used, the Leader's share is structured as follows:
struct {
Prio3Field meas_share[F * prio3.flp.MEAS_LEN];
Prio3Field proofs_share[F * prio3.flp.PROOF_LEN * prio3.PROOFS];
} Prio3LeaderShare;
When joint randomness is not used, the Helpers' shares are structured
as follows:
struct {
Prio3Seed share;
} Prio3HelperShare;
When joint randomness is used, the Leader's input share is structured as
follows:
struct {
Prio3LeaderShare inner;
Prio3Seed blind;
} Prio3LeaderShareWithJointRand;
Finally, when joint randomness is used, the Helpers' shares are structured as
follows:
struct {
Prio3HelperShare inner;
Prio3Seed blind;
} Prio3HelperShareWithJointRand;
7.2.7.3.
Verifier Share
When joint randomness is not used, the verifier share is structured as follows:
struct {
Prio3Field verifiers_share[F * V];
} Prio3VerifierShare;
where
V = prio3.flp.VERIFIER_LEN * prio3.PROOFS
. When joint randomness is
used, the verifier share includes the Aggregator's joint randomness part and is
structured as follows:
struct {
Prio3Field verifiers_share[F * V];
Prio3Seed joint_rand_part;
} Prio3VerifierhareWithJointRand;
7.2.7.4.
Verifier Message
When joint randomness is not used, the verifier message is the empty string.
Otherwise the verifier message consists of the joint randomness seed computed
by the Aggregators:
struct {
Prio3Seed joint_rand;
} Prio3VerifierMessageWithJointRand;
7.2.7.5.
Aggregation
Aggregate shares are structured as follows:
struct {
Prio3Field agg_share[F * prio3.flp.OUTPUT_LEN];
} Prio3AggShare;
7.3.
FLP Specification
This section specifies an implementation of the
Flp
interface (
Section 7.1
) based
on the construction from
BBCGGI19
, Section 4.2. The types and parameters
required by this interface are listed in the table below.
Section 7.3.1
provides an overview of the proof system and some
extensions to it.
Section 7.3.2
defines validity circuits, the core
component of the proof system that determines measurement validity and how
measurements are aggregated. The proof-generation algorithm, query algorithm,
and decision algorithm are defined in
Section 7.3.3
Section 7.3.4
, and
Section 7.3.5
respectively.
Table 8
FLP parameters for a validity circuit.
Parameter
Value
valid
An instance of
Valid
Section 7.3.2
).
field
valid.field
PROVE_RAND_LEN
valid.prove_rand_len()
QUERY_RAND_LEN
valid.query_rand_len()
JOINT_RAND_LEN
valid.JOINT_RAND_LEN
MEAS_LEN
valid.MEAS_LEN
OUTPUT_LEN
valid.OUTPUT_LEN
PROOF_LEN
valid.proof_len()
VERIFIER_LEN
valid.verifier_len()
Measurement
As defined by
valid
AggResult
As defined by
valid
7.3.1.
Overview
An FLP is a type of "zero-knowledge proof". A conventional zero-knowledge proof
system involves two parties:
The prover, who holds a measurement and generates a proof of the
measurement's validity
The verifier who holds an encryption of, or commitment to, the measurement
and checks the proof
The proof system here is much the same, except the verifier is split across
multiple Aggregators, each of which has a secret share of the measurement
rather than a commitment to it.
Validity is defined in terms of an arithmetic circuit evaluated over the
measurement. The inputs to this circuit are elements of a finite field that
comprise the encoded measurement; the gates of the circuit are multiplication,
addition, and subtraction operations; and the output of the circuit is a single
field element. If the value is zero, then the measurement is deemed valid;
otherwise, if the output is non-zero, then the measurement is deemed invalid.
For example, the simplest circuit specified in this document is the following
Section 7.4.1
):
C(x) = x * (x-1)
This circuit contains one subtraction gate (
x-1
) and one multiplication gate
x * (x-1)
). Observe that
C(x) = 0
if and only if
is in the range
[0, 2)
The goal of the proof system is to allow each Aggregator to privately and
correctly compute a share of
C(x)
from its share of
. Then all they need
to do to determine validity is to broadcast their shares of
C(x)
Suppose for a moment that
is an affine arithmetic circuit, meaning its only
operations are addition, subtraction, and multiplication-by-constant. (The
circuit above is non-affine because it contains a multiplication gate with two
non-constant inputs.) Then each Aggregator can compute its share locally, since
C(x_shares[0] + ... + x_shares[SHARES-1]) =
C(x_shares[0]) + ... + C(x_shares[SHARES-1])
(Note that, for this equality to hold, it is necessary to scale any addition of
a constant in the circuit by
1/SHARES
.) However, this is not the case if
contains multiplication gates with two non-constant inputs. Thus the goal is to
transform these multiplication gates into computations on secret shared data
that each Aggregator can perform locally.
The key idea is to have the prover construct a polynomial
such that
p(j)
is equal to the output of the
-th multiplication gate. Polynomial evaluation
is fully linear, which means the polynomial can be secret
shared in a way that allows each Aggregator to compute a share of
p(j)
for
any
. These intermediate results can then be combined with the affine
arithmetic operations of the validity circuit to produce the final output.
Applying this idea to the example circuit
above:
The Client, given its measurement
, constructs the lowest degree
polynomial
for which
p(0) = s
and
p(1) = x * (x-1)
, where
is a
random blinding value generated by the Client. (The blinding value is to
protect the privacy of the measurement.) It then sends shares of
and
shares of
to each of the Aggregators.
Each Aggregator locally computes and broadcasts its share of
p(1)
, which
is equal to its share of
C(x)
In fact, the FLP is slightly more general than this. One can replace the
multiplication gate with any non-affine sub-circuit and apply the same idea.
For example, in
Section 7.4.2
, the validity circuit uses the following
sub-circuit multiple times:
Range2(x) = x * (x-1) = x**2 - x
(This is the same functionality computed by the example circuit
above.)
Here again one can interpolate the lowest degree polynomial
for which
p(j)
is the value of the
-th call to
Range2
in the validity circuit. Each
validity circuit defines a sub-circuit that encapsulates its non-affine
arithmetic operations. This sub-circuit is called the "gadget".
Finally, the proof system has one more important component. It is possible for
a malicious Client to produce a gadget polynomial
that would result in
C(x)
being computed incorrectly, potentially resulting in an invalid
measurement being accepted. To prevent this, the Aggregators perform a
probabilistic test to check that the gadget polynomial was constructed
properly. This "gadget test", and the procedure for constructing the
polynomial, are described in detail in
Section 7.3.3
7.3.1.1.
Extensions
The FLP described in
Section 7.3
extends the proof system of
BBCGGI19
, Section 4.2 in a few ways.
First, the validity circuit in the construction includes an additional, random
input (this is the "joint randomness" derived from the measurement shares in
Prio3; see
Section 7.2
). This allows for circuit optimizations that
trade a small soundness error for a shorter proof. For example, consider a
circuit that recognizes the set of length-
vectors for which each element is
either one or zero. A deterministic circuit could be constructed for this
language, but it would involve a large number of multiplications that would
result in a large proof. (See the discussion in
BBCGGI19
, Section 5.2 for
details). A much shorter proof can be constructed for the following randomized
circuit:
C(x, r) = r * Range2(x[0]) + ... + r**N * Range2(x[N-1])
(Note that this is a special case of
BBCGGI19
, Theorem 5.2.) Here
is
the length-
input and
is a random field element. The gadget circuit
Range2
is the "range-check" polynomial described above, i.e.,
Range2(x) =
x**2 - x
. The idea is that, if
is valid, i.e., each
x[j]
is in
the range
[0, 2)
, then the circuit will evaluate to zero regardless of the
value of
; but if some
x[j]
is not in the range
[0, 2)
, then the output
will be non-zero with high probability.
The second extension implemented by the FLP allows the validity circuit to
contain multiple gadget types. (This generalization was suggested in
BBCGGI19
, Remark 4.5.) This provides additional flexibility for designing
circuits by allowing multiple, non-affine sub-circuits. For example, the
following circuit is allowed:
C(x, r) = r * Range2(x[0]) + ... + r**L * Range2(x[L-1]) + \
r**(L+1) * Range3(x[L]) + ... + r**N * Range3(x[N-1])
where
Range3(x) = x**3 - 3x**2 + 2x
. This circuit checks that the first
inputs are in the range
[0, 2)
and the last
N-L
inputs are in the range
[0, 3)
. The same circuit can be expressed using a simpler gadget, namely
multiplication, but the resulting proof would be longer.
Third, rather than defining polynomials at inputs
, ...,
where
is the
-th invocation of the gadget, the roots of unity for
the field are used. This allows using the NTT algorithm for efficiency.
Note that the roots of unity are powers of the generator for the
NTT-friendly field (see
Section 6.1.2
).
Finally, the validity circuit in the FLP may have any number of outputs (at
least one). The input is said to be valid if each of the outputs is zero. To
save bandwidth, the FLP takes a random linear combination of the outputs. If
each of the outputs is zero, then the reduced output will be zero; but if one
of the outputs is non-zero, then the reduced output will be non-zero with high
probability.
7.3.2.
Validity Circuits
An instance of the proof system is defined in terms of a validity circuit that
implements the
Valid
interface specified in this section. The parameters are
listed in the table below.
Table 9
Validity circuit parameters.
Parameter
Description
GADGETS: list[Gadget]
A list of gadgets.
GADGET_CALLS: list[int]
Number of times each gadget is called.
MEAS_LEN: int
Length of the measurement.
JOINT_RAND_LEN: int
Length of the joint randomness.
EVAL_OUTPUT_LEN: int
Length of the circuit output.
OUTPUT_LEN: int
Length of the aggregatable output.
Measurement
Type of the measurement.
AggResult
Type of the aggregate result.
field: type[F]
Class object for the field (
Section 6.1.2
The circuit is invoked with the following method:
valid.eval(meas: list[F], joint_rand: list[F], num_shares: int) -> list[F]
evaluates the arithmetic circuit on a measurement and joint randomness. The
output is a list of field elements: if every element is equal to
valid.field(0)
, then the circuit is said to "accept" the measurement;
otherwise, if any element is not equal to
valid.field(0)
, then the circuit
is said to "reject" the measurement.
This method can also be called on a secret share of the measurement, in which
case it produces a secret share of the output.
The circuit must be composed of affine gates and gadget calls, so that the
verifier may check the prover's proof and circuit evaluation using linear
queries. This means that all non-affine multiplications in the circuit must
be encapsulated in gadget calls. Additions of constants must be rescaled by
the inverse of
num_shares
Pre-conditions:
The length of
meas
MUST
be
valid.MEAS_LEN
The length of
joint_rand
MUST
be
valid.JOINT_RAND_LEN
num_shares
MUST
be the number of secret shares of
meas
, or
if
meas
is not secret shared.
Post-conditions:
The length of the output
MUST
be
valid.EVAL_OUTPUT_LEN
Each circuit has a list of gadgets, denoted
GADGETS
, that are invoked by
valid.eval()
. The circuit evaluated by the gadget should be non-affine, and
MUST
be arithmetic, i.e., composed only of multiplication, addition, and subtraction
gates. An instance of class
Gadget
has the following interface:
ARITY: int
is the number of input wires. For example, the multiplication
gadget
Mul(x,y) = x*y
has arity of 2.
DEGREE: int
is the arithmetic degree of the gadget circuit. This is defined
to be the degree of the polynomial that computes it. This exists
because the circuit is arithmetic. For example,
Mul
has degree 2.
gadget.eval(field: type[F], inp: list[F]) -> F
evaluates the gadget over
the given inputs and field.
gadget.eval_poly(field: type[F], inp_poly: list[list[F]]) -> list[F]
is the
same as
gadget.eval()
except it evaluates the circuit over the polynomial
ring of the field. This is well defined because the circuit is arithmetic.
In addition to the list of gadgets, the validity circuit specifies how many
times each gadget is called (
GADGET_CALLS
). The circuit needs to define an
ordering of the calls it makes to each gadget, so that all parties agree on how
to identify recorded wire values. It also specifies the length of the circuit's
input (
MEAS_LEN
), the length of the joint randomness (
JOINT_RAND_LEN
), and
the length of the circuit's output (
EVAL_OUTPUT_LEN
).
A validity circuit also specifies parameters and methods needed for Prio3
aggregation. These are used to implement the interface in
Section 7.1.1
valid.encode(measurement: Measurement) -> list[F]
returns a vector of
length
MEAS_LEN
representing a measurement of type
Measurement
valid.truncate(meas: list[F]) -> list[F]
returns a vector of length
OUTPUT_LEN
representing (a share of) an aggregatable output.
valid.decode(agg: list[F], num_measurements: int) -> AggResult
returns an
aggregate result of type
AggResult
. This computation may depend on the
number of outputs aggregated.
Finally, the following are helper methods used to instantiate parameters of the
Flp
interface (
Section 7.1
):
def prove_rand_len(self) -> int:
"""Length of the prover randomness."""
return sum(g.ARITY for g in self.GADGETS)
def query_rand_len(self) -> int:
"""Length of the query randomness."""
query_rand_len = len(self.GADGETS)
if self.EVAL_OUTPUT_LEN > 1:
query_rand_len += self.EVAL_OUTPUT_LEN
return query_rand_len
def proof_len(self) -> int:
"""Length of the proof."""
length = 0
for (g, g_calls) in zip(self.GADGETS, self.GADGET_CALLS):
p = wire_poly_len(g_calls)
length += g.ARITY + gadget_poly_len(g.DEGREE, p)
return length
def verifier_len(self) -> int:
"""Length of the verifier message."""
length = 1
for g in self.GADGETS:
length += g.ARITY + 1
return length
def wire_poly_len(gadget_calls: int) -> int:
"""
Calculates the number of coordinates in each wire polynomial for
a gadget.
"""
return next_power_of_2(1 + gadget_calls)
def gadget_poly_len(gadget_degree: int,
wire_polynomial_len: int) -> int:
"""
Calculates the number of coordinates in a gadget polynomial.
"""
return gadget_degree * (wire_polynomial_len - 1) + 1
7.3.3.
Generating the Proof
Figure 9
Components of the proof generation algorithm.
The proof generation algorithm invokes the validity circuit on the encoded
measurement and joint randomness. The validity circuit in turn invokes the
gadgets defined by the circuit. The prover records the values on input wires of
gadget instances during circuit evaluation, and constructs gadget polynomials
that the verifier will use to compute the outputs of each gadget. Additionally,
the prove randomness is used as a blinding factor when constructing gadget
polynomials.
To generate the gadget polynomials, the prover evaluates the validity circuit,
and records the values on each input wire of each call to each gadget. This is
accomplished by "wrapping" each gadget in a class
ProveGadget
that records
the wire inputs. This class is listed in
Appendix A.4
. Denote the value
of the
-th wire for the
-th invocation of gadget
as
g.wires[j][k]
Next, the prover computes each of the "wire polynomials" for each gadget. For
each wire polynomial, take one prove randomness value and designate it the
"wire seed" for that polynomial. The
-th wire polynomial is the lowest
degree polynomial that evaluates to the "wire seed" at one point and
g.wires[j][k]
at a sequence of other points. The gadget polynomial is
obtained by evaluating the gadget on the wire polynomials.
def prove(self,
meas: list[F],
prove_rand: list[F],
joint_rand: list[F]) -> list[F]:
# Evaluate the validity circuit, recording the value of each
# input wire for each evaluation of each gadget.
valid = ProveGadget.wrap(self.valid, prove_rand)
valid.eval(meas, joint_rand, 1)
# Construct the proof, which consists of the wire seeds and
# gadget polynomial for each gadget.
proof = []
for g, g_calls in zip(valid.GADGETS, valid.GADGET_CALLS):
g = cast(ProveGadget[F], g)
# Define the wire polynomial length `p` as the smallest power
# of two accommodating all gadget calls plus one constant
# term for the seed.
p = wire_poly_len(g_calls)
# The validity circuit evaluation defines one polynomial for
# each input wire of each gadget.
# For each wire `j`, the vector `g.wires[j]` of length `p`
# is built as follows:
# - `g.wires[j][0]` is set to the seed for wire `j` (from
# the prover's randomness).
# - The subsequent entries are the assigned values from each
# gadget call.
# - Pad the vector with zeros to reach length `p`.
# The wire polynomial is then defined by its evaluations:
# `wire_poly(alpha**k) = g.wires[j][k]`
# for all `k`, where `alpha` is a `p`-th root of unity.
wire_polys = [g.wires[j] for j in range(g.ARITY)]
wire_seeds = [g.wires[j][0] for j in range(g.ARITY)]
proof += wire_seeds
# Compute the gadget polynomial by evaluating the gadget
# on the wire polynomials. By construction we have that
# `gadget_poly(alpha**k)` is the `k`-th output.
gadget_poly = g.eval_poly(self.field, wire_polys)
proof += gadget_poly[:gadget_poly_len(g.DEGREE, p)]
return proof
7.3.4.
Querying the Proof
Figure 10
Components of the query algorithm.
The query algorithm invokes the validity circuit on the encoded measurement and
joint randomness. It evaluates the gadget polynomials encoded by the proof
(share) to produce (a share of) each gadget output. The verifier (share)
consists of (a share of) the validity circuit's output and (a share of) each
gadget test. The gadget tests consume the query randomness.
The goal of each gadget test is to ensure the inputs used by the prover to
generate the gadget polynomial match the inputs used to evaluate it. This is
done by partially reconstructing the gadget polynomial and evaluating it at a
random point: when the gadget polynomial is evaluated at the same point, the
result should be the same.
To start a gadget test, first construct the (shares of the) wire polynomials
just as the prover did. Then record the input (share) of the
-th wire of the
-th invocation of the gadget as
g.wires[j][k]
. Again, this is accomplished
by a wrapper gadget,
QueryGadget
, listed in
Appendix A.4
. This gadget
also evaluates the gadget polynomial for each gadget invocation in order to
produce the gadget's output. Then compute the wire polynomials from the
recorded values.
Note that these polynomials are in the Lagrange basis representation
by construction, and the gadget polynomials parsed from the proof share
are similarly transmitted in the Lagrange basis representation.
Generating additive secret shares of a polynomial works equally both in
the monomial and the Lagrange basis (
Section 6.1.3
).
Next, choose a random point
(parsed from the query randomness), evaluate
each wire polynomial at
, and evaluate the gadget polynomial at
Polynomial evaluation in the Lagrange basis is
faster by using the algorithm by
Faz25
without resorting to
polynomial interpolation.
The results are recorded in the verifier message passed to the decision
algorithm, where the test is finished.
The random point
MUST NOT
be one of the fixed evaluation points used to
define the wire polynomials. Otherwise, the verifier message may partially
leak the encoded measurement.
def query(self,
meas: list[F],
proof: list[F],
query_rand: list[F],
joint_rand: list[F],
num_shares: int) -> list[F]:
# Evaluate the validity circuit, recording the value of each
# input wire for each evaluation of each gadget. Use the gadget
# polynomials encoded by `proof` to compute the gadget outputs.
valid = QueryGadget.wrap(self.valid, proof)
out = valid.eval(meas, joint_rand, num_shares)
# Reduce the output.
if self.valid.EVAL_OUTPUT_LEN > 1:
(rand, query_rand) = front(
self.valid.EVAL_OUTPUT_LEN,
query_rand,
v = self.field(0)
for (r, out_elem) in zip(rand, out):
v += r * out_elem
else:
[v] = out
# Construct the verifier message, which consists of the reduced
# circuit output and each gadget test.
lag = Lagrange(self.field)
verifier = [v]
for (g, t) in zip(cast(list[QueryGadget[F]], valid.GADGETS),
query_rand):
p = len(g.wires[0])
# Abort if `t` is one of the inputs used to compute the wire
# polynomials so that the verifier message doesn't leak the
# gadget output. It suffices to check if `t` is a root of
# unity, which implies it is a power of `alpha`.
if t ** p == self.field(1):
raise ValueError('test point is a root of unity')
# To test the gadget, we re-compute the wire polynomials and
# check for consistency with the gadget polynomial provided
# by the prover. To start, evaluate the gadget polynomial
# and each of the wire polynomials at the random point `t`.
wire_checks = lag.poly_eval_batched(g.wires[:g.ARITY], t)
gadget_check = lag.poly_eval(g.poly, t)
verifier += wire_checks
verifier.append(gadget_check)
return verifier
7.3.5.
Deciding Validity
Figure 11
Components of the decision algorithm.
The decision algorithm consumes the verifier message. (Each of the Aggregators
computes an additive share of the verifier message after the previous step.) The
verifier message consists of the reduced circuit output and the gadget tests.
To finish each gadget test, evaluate the gadget on the wire checks: if the
encoded measurement and joint randomness used to generate the proof are the
same as the measurement (share) and joint randomness used to verify the proof,
then the output of the gadget will be equal to the gadget check; otherwise, the
output will not equal the gadget check with high probability.
def decide(self, verifier: list[F]) -> bool:
# Check the output of the validity circuit.
([v], verifier) = front(1, verifier)
if v != self.field(0):
return False
# Complete each gadget test.
for g in self.valid.GADGETS:
(wire_checks, verifier) = front(g.ARITY, verifier)
([gadget_check], verifier) = front(1, verifier)
if g.eval(self.field, wire_checks) != gadget_check:
return False
return True
7.4.
Variants
This section specifies instantiations of Prio3 for various aggregation tasks.
Each variant is determined by a field (
Section 6.1
), a validity circuit
Section 7.3.2
),and the number of proofs to generate and verify. All
gadgets are listed in
Appendix A
. Test vectors for each can be found in
Appendix C
7.4.1.
Prio3Count
Table 10
Parameters for Prio3Count.
Parameter
Value
field
Field64
Table 4
Valid
Count(field)
(this section)
PROOFS
The first variant of Prio3 is for a simple counter: each measurement is either
one or zero and the aggregate result is the sum of the measurements. Its
validity circuit uses the multiplication gadget
Mul
specified in
Appendix A.1
, which takes two inputs and multiplies them. The circuit is
specified below:
class Count(Valid[int, int, F]):
GADGETS: list[Gadget[F]] = [Mul()]
GADGET_CALLS = [1]
MEAS_LEN = 1
JOINT_RAND_LEN = 0
OUTPUT_LEN = 1
EVAL_OUTPUT_LEN = 1
# Class object for the field.
field: type[F]
def __init__(self, field: type[F]):
self.field = field
def encode(self, measurement: int) -> list[F]:
return [self.field(measurement)]
def eval(
self,
meas: list[F],
joint_rand: list[F],
_num_shares: int) -> list[F]:
squared = self.GADGETS[0].eval(self.field,
[meas[0], meas[0]])
return [squared - meas[0]]
def truncate(self, meas: list[F]) -> list[F]:
return meas
def decode(self, output: list[F], _num_measurements: int) -> int:
return output[0].int()
7.4.2.
Prio3Sum
Table 11
Parameters for Prio3Sum.
Parameter
Value
field
Field64
Table 4
Valid
Sum(field, max_measurement)
(this section)
PROOFS
The next variant of Prio3 supports summing of integers in a pre-determined
range. Each measurement is an integer in the range
[0, max_measurement]
where
max_measurement
defines the largest valid measurement.
The range check is accomplished by encoding the measurement as a vector of field
elements with value zero or one, such that a weighted sum of these values can only
be in the range
[0, max_measurement]
All but the last of the weights are successive powers of two, as in
the binary bit decomposition, and the last weight is chosen such that the sum of
all weights is equal to
max_measurement
. With these weights, valid
measurements have either one or two possible representations as vectors of
field elements with value zero or one, and invalid measurements cannot be
represented.
The validity circuit checks that each entry of the bit vector has a value of
zero or one. It uses the polynomial-evaluation gadget
PolyEval
specified in
Appendix A.2
. The polynomial is
p(x) = x**2 - x
, which is equal to
if and only if
is in the range
[0, 2)
. The complete circuit is specified
below:
Note that decoding a sequence of "bits" into an integer is a linear operation,
specifically, a linear combination with the constant weights. Thus, decoding
secret shares of a bit-encoded integer will produce secret shares of the
original integer.
class Sum(Valid[int, int, F]):
JOINT_RAND_LEN = 0
OUTPUT_LEN = 1
field: type[F]
def __init__(self, field: type[F], max_measurement: int):
self.field = field
bits = max_measurement.bit_length()
self.bits = bits
self.max_measurement = max_measurement
self.GADGET_CALLS = [self.bits]
self.GADGETS = [PolyEval([0, -1, 1], self.bits)]
self.MEAS_LEN = self.bits
self.EVAL_OUTPUT_LEN = self.bits
def encode(self, measurement: int) -> list[F]:
return encode_range_checked_int(
self.field,
measurement,
self.max_measurement,
def eval(
self,
meas: list[F],
joint_rand: list[F],
_num_shares: int) -> list[F]:
out = []
for b in meas:
out.append(self.GADGETS[0].eval(self.field, [b]))
return out
def truncate(self, meas: list[F]) -> list[F]:
return [
decode_range_checked_int(
self.field,
meas,
self.max_measurement,
def decode(self, output: list[F], _num_measurements: int) -> int:
return output[0].int()
def encode_range_checked_int(
field: type[F],
value: int,
max_measurement: int) -> list[F]:
"""
Encode an integer into multiple field elements, representing a
weighted sum of zeros and ones, to allow for efficient range
proofs.
Pre-conditions:
- `value >= 0`
- `max_measurement > 0`
- `value <= max_measurement`
- `value < field.MODULUS`
- `max_measurement < field.MODULUS`
"""
if value > max_measurement:
raise ValueError("measurement is too large")
bits = max_measurement.bit_length()
rest_all_ones_value = 2 ** (bits - 1) - 1
last_weight = max_measurement - rest_all_ones_value
# Implementation note: this conditional should be replaced with
# constant time operations in order to reduce leakage via timing
# side channels.
if value <= rest_all_ones_value:
rest = value
last_elem = field(0)
else:
rest = value - last_weight
last_elem = field(1)
encoded = []
for l in range(bits - 1):
encoded.append(field((rest >> l) & 1))
encoded.append(last_elem)
return encoded
def decode_range_checked_int(
field: type[F],
encoded: list[F],
max_measurement: int) -> F:
"""
Decode a field element from a vector of field elements produced
by `encode_range_checked_int()`.
This may also be applied to secret shares of an encoded integer,
since it is a linear function.
"""
bits = max_measurement.bit_length()
rest_all_ones_value = 2 ** (bits - 1) - 1
last_weight = max_measurement - rest_all_ones_value
decoded = field(0)
for (l, bit) in enumerate(encoded[:bits - 1]):
decoded += field(1 << l) * bit
decoded += field(last_weight) * encoded[bits - 1]
return decoded
7.4.3.
Prio3SumVec
Table 12
Parameters for Prio3SumVec.
Parameter
Value
field
Field128
Table 4
Valid
SumVec(field, length, max_measurement, chunk_length)
(this section)
PROOFS
This instance of Prio3 supports summing vectors of integers. It has three
parameters:
length
max_measurement
, and
chunk_length
. Each measurement is
a vector of positive integers with length equal to the
length
parameter. Each
element of the measurement is an integer in the range
[0, max_measurement]
Let
bits = max_measurement.bit_length()
, the number of bits needed to encode
the largest valid measurement.
It s
RECOMMENDED
to set
chunk_length
to an integer near the square root of
length * bits
, (see
Section 7.4.3.1
).
The circuit is denoted
SumVec
. Each measurement is encoded as a vector of
field elements with a length of
length * bits
. The field elements in the
encoded vector represent modified bit decompositions of each of the measurement
vector's elements, following the
Sum
circuit's encoding procedure
Section 7.4.2
), concatenated together.
The validity circuit uses the
ParallelSum
gadget in
Appendix A.3
This gadget applies an arithmetic subcircuit to multiple inputs in parallel,
then sums the results. Along with the subcircuit, the
parallel-sum gadget is parameterized by an integer, denoted
count
, specifying
how many times to call the subcircuit. It takes in a list of inputs and passes
them through to instances of the subcircuit in the same order. It returns the
sum of the subcircuit outputs.
Note that only the
ParallelSum
gadget itself, and not its subcircuit,
participates in the FLP's wire recording during evaluation, gadget consistency
proofs, and proof validation, even though the subcircuit is provided to
ParallelSum
gadget as an implementation of the
Gadget
interface.
The
SumVec
validity circuit checks that the encoded measurement consists of
ones and zeros. Rather than use the
PolyEval
gadget on each element, as in
the
Sum
validity circuit, it instead uses
Mul
subcircuits (
Appendix A.1
and "free" constant multiplication and addition gates to simultaneously
evaluate the same range check polynomial on each element, and multiply by a
constant. One of the two
Mul
subcircuit inputs is equal to a measurement
element multiplied by a power of one of the elements of the joint randomness
vector, and the other is equal to the same measurement element minus one. These
Mul
subcircuits are evaluated by a
ParallelSum
gadget, and the results are
added up both within the
ParallelSum
gadget and after it.
The complete circuit is specified below:
class SumVec(Valid[list[int], list[int], F]):
EVAL_OUTPUT_LEN = 1
length: int
max_measurement: int
chunk_length: int
field: type[F]
def __init__(self,
field: type[F],
length: int,
max_measurement: int,
chunk_length: int):
"""
Instantiate the `SumVec` circuit for measurements with
`length` elements, each in the range `[0, max_measurement]`.
"""
self.field = field
self.length = length
bits = max_measurement.bit_length()
self.bits = bits
self.max_measurement = max_measurement
self.chunk_length = chunk_length
self.GADGETS = [ParallelSum(Mul(), chunk_length)]
self.GADGET_CALLS = [
(length * bits + chunk_length - 1) // chunk_length
self.MEAS_LEN = length * bits
self.OUTPUT_LEN = length
self.JOINT_RAND_LEN = self.GADGET_CALLS[0]
def encode(self, measurement: list[int]) -> list[F]:
encoded = []
for val in measurement:
encoded += encode_range_checked_int(
self.field,
val,
self.max_measurement,
return encoded
def eval(
self,
meas: list[F],
joint_rand: list[F],
num_shares: int) -> list[F]:
out = self.field(0)
shares_inv = self.field(num_shares).inv()
for i in range(self.GADGET_CALLS[0]):
r = joint_rand[i]
r_power = r
inputs: list[Optional[F]]
inputs = [None] * (2 * self.chunk_length)
for j in range(self.chunk_length):
index = i * self.chunk_length + j
if index < len(meas):
meas_elem = meas[index]
else:
meas_elem = self.field(0)
inputs[j * 2] = r_power * meas_elem
inputs[j * 2 + 1] = meas_elem - shares_inv
r_power *= r
out += self.GADGETS[0].eval(
self.field,
cast(list[F], inputs),
return [out]
def truncate(self, meas: list[F]) -> list[F]:
truncated = []
for i in range(self.length):
truncated.append(
decode_range_checked_int(
self.field,
meas[i * self.bits:(i + 1) * self.bits],
self.max_measurement,
return truncated
def decode(
self,
output: list[F],
_num_measurements: int) -> list[int]:
return [x.int() for x in output]
7.4.3.1.
Selection of
ParallelSum
Chunk Length
The
chunk_length
parameter provides a trade-off between the arity of the
ParallelSum
gadget (
Appendix A.3
) and the number of times the
gadget is called. The proof length is asymptotically minimized when the chunk
length is near the square root of the length of the encoded measurement.
However, the relationship between VDAF parameters and proof length is
complicated, involving two forms of rounding:
The circuit pads the inputs to its last
ParallelSum
gadget call, up to
the chunk length.
The proof system rounds the degree of wire polynomials, determined by the
number of times a gadget is called, up to the next power of two.
Therefore, the optimal choice of
chunk_length
for a concrete measurement size
will vary, and must be found through trial and error. Setting
chunk_length
equal to the square root of the appropriate measurement length will result in
proofs up to 50% larger than the optimal proof size.
7.4.4.
Prio3Histogram
Table 13
Parameters for Prio3Histogram.
Parameter
Value
field
Field128
Table 4
Valid
Histogram(field, length, chunk_length)
(this section)
PROOFS
This variant of Prio3 allows for estimating the distribution of some quantity
by computing a simple histogram. Each measurement increments one histogram
bucket, out of a set of fixed buckets. (Bucket indexing begins at
.) For
example, the buckets might quantize the real numbers, and each measurement
would report the bucket that the corresponding client's real-numbered value
falls into. The aggregate result counts the number of measurements in each
bucket.
The validity circuit is denoted
Histogram
. It has two parameters:
length
the number of histogram buckets; and
chunk_length
, which is used by a
circuit optimization described below. It is
RECOMMENDED
to set
chunk_length
to an integer near the square root of
length
(see
Section 7.4.3.1
).
The measurement is encoded as a one-hot vector representing the bucket into
which the measurement falls. The circuit checks for one-hotness in two steps,
by checking that the encoded measurement consists of ones and zeros, and by
checking that the sum of all elements in the encoded measurement is equal to
one. The individual checks constitute the output of the circuit.
As in the
SumVec
validity circuit (
Section 7.4.3
), the first part of the
validity circuit uses the
ParallelSum
Appendix A.3
) gadget to
perform range checks while achieving a smaller proof size. The
ParallelSum
gadget uses
Mul
subcircuits (
Appendix A.1
) to evaluate a range check
polynomial on each element, and includes an additional constant multiplication.
One of the two
Mul
subcircuit inputs is equal to a measurement element
multiplied by a power of an element of the joint randomness vector, and the
other is equal to the same measurement element minus one. The results are added
up both within the
ParallelSum
gadget and after it.
class Histogram(Valid[int, list[int], F]):
EVAL_OUTPUT_LEN = 2
field: type[F]
length: int
chunk_length: int
def __init__(self,
field: type[F],
length: int,
chunk_length: int):
"""
Instantiate an instance of the `Histogram` circuit with the
given `length` and `chunk_length`.
"""
self.field = field
self.length = length
self.chunk_length = chunk_length
self.GADGETS = [ParallelSum(Mul(), chunk_length)]
self.GADGET_CALLS = [
(length + chunk_length - 1) // chunk_length]
self.MEAS_LEN = self.length
self.OUTPUT_LEN = self.length
self.JOINT_RAND_LEN = self.GADGET_CALLS[0]
def encode(self, measurement: int) -> list[F]:
encoded = [self.field(0)] * self.length
encoded[measurement] = self.field(1)
return encoded
def eval(
self,
meas: list[F],
joint_rand: list[F],
num_shares: int) -> list[F]:
# Check that each bucket is one or zero.
range_check = self.field(0)
shares_inv = self.field(num_shares).inv()
for i in range(self.GADGET_CALLS[0]):
r = joint_rand[i]
r_power = r
inputs: list[Optional[F]]
inputs = [None] * (2 * self.chunk_length)
for j in range(self.chunk_length):
index = i * self.chunk_length + j
if index < len(meas):
meas_elem = meas[index]
else:
meas_elem = self.field(0)
inputs[j * 2] = r_power * meas_elem
inputs[j * 2 + 1] = meas_elem - shares_inv
r_power *= r
range_check += self.GADGETS[0].eval(
self.field,
cast(list[F], inputs),
# Check that the buckets sum to 1.
sum_check = -shares_inv
for b in meas:
sum_check += b
return [range_check, sum_check]
def truncate(self, meas: list[F]) -> list[F]:
return meas
def decode(
self,
output: list[F],
_num_measurements: int) -> list[int]:
return [bucket_count.int()
for bucket_count in output]
7.4.5.
Prio3MultihotCountVec
Table 14
Parameters for Prio3MultihotCountVec.
Parameter
Value
field
Field128
Table 4
Valid
MultihotCountVec(field, length, max_weight, chunk_length)
(this section)
PROOFS
For this instance of Prio3, each measurement is a vector of Boolean
values, where the number of
True
values is bounded. This provides a
functionality similar to Prio3Histogram except that more than one entry (or none
at all) may be non-zero. This allows Prio3MultihotCountVec to be composed with a
randomized response mechanism, like
EPK14
, for providing differential
privacy. (For example, each Client would set each entry with some small
probability.)
The validity circuit is denoted
MultihotCountVec
and has three parameters:
length
, the number of entries in the count vector;
max_weight
, the maximum
number of
True
entries (i.e., the weight must be at most
max_weight
); and
chunk_length
, used the same way as in
Section 7.4.3
and
Section 7.4.4
The Client's encoded measurement includes both the count vector and the
weight of the count vector. The weight is encoded as multiple field
elements, using the same modified bit decomposition as in the
Sum
circuit's encoding procedure (
Section 7.4.2
).
The verifier checks that each entry of the encoded measurement is a bit (i.e.,
either one or zero), including both the count vector and the encoding of the
reported weight. It then decodes the reported weight and subtracts it from
sum(count_vec)
, where
count_vec
is the count vector. The result is
zero if and only if the reported weight is equal to the true weight. The two
checks constitute the output of the circuit. The complete circuit is defined
below.
class MultihotCountVec(Valid[list[bool], list[int], F]):
EVAL_OUTPUT_LEN = 2
field: type[F]
def __init__(self,
field: type[F],
length: int,
max_weight: int,
chunk_length: int):
"""
Instantiate an instance of the this circuit with the given
`length`, `max_weight`, and `chunk_length`.
Pre-conditions:
- `length > 0`
- `0 < max_weight` and `max_weight <= length`
- `chunk_length > 0`
"""
self.field = field
# Compute the number of bits to represent `max_weight`.
self.bits_for_weight = max_weight.bit_length()
# Make sure `length` and `max_weight` don't overflow the
# field modulus. Otherwise we may not correctly compute the
# sum of measurement vector entries during circuit evaluation.
if self.field.MODULUS <= length:
raise ValueError('length is too large for the '
'current field size')
if self.field.MODULUS <= max_weight:
raise ValueError('max_weight is too large for the '
'current field size')
self.length = length
self.max_weight = max_weight
self.chunk_length = chunk_length
self.GADGETS: list[Gadget[F]] = [
ParallelSum(Mul(), chunk_length),
self.GADGET_CALLS = [
(length + self.bits_for_weight + chunk_length - 1)
// chunk_length
self.MEAS_LEN = self.length + self.bits_for_weight
self.OUTPUT_LEN = self.length
self.JOINT_RAND_LEN = self.GADGET_CALLS[0]
def encode(self, measurement: list[bool]) -> list[F]:
if len(measurement) != self.length:
raise ValueError('invalid Client measurement length')
# The first part is the vector of counters.
count_vec = [self.field(int(x)) for x in measurement]
# The second part is the reported weight.
weight_reported = sum(measurement)
encoded = []
encoded += count_vec
encoded += encode_range_checked_int(
self.field,
weight_reported,
self.max_weight,
return encoded
def eval(
self,
meas: list[F],
joint_rand: list[F],
num_shares: int) -> list[F]:
# Check that each entry in the input vector is one or zero.
range_check = self.field(0)
shares_inv = self.field(num_shares).inv()
for i in range(self.GADGET_CALLS[0]):
r = joint_rand[i]
r_power = r
inputs: list[Optional[F]]
inputs = [None] * (2 * self.chunk_length)
for j in range(self.chunk_length):
index = i * self.chunk_length + j
if index < len(meas):
meas_elem = meas[index]
else:
meas_elem = self.field(0)
inputs[j * 2] = r_power * meas_elem
inputs[j * 2 + 1] = meas_elem - shares_inv
r_power *= r
range_check += self.GADGETS[0].eval(
self.field,
cast(list[F], inputs),
# Check that the sum of the counters is equal to the value
# claimed by the Client.
count_vec = meas[:self.length]
weight = sum(count_vec, self.field(0))
weight_reported = decode_range_checked_int(
self.field,
meas[self.length:],
self.max_weight,
weight_check = weight - weight_reported
return [range_check, weight_check]
def truncate(self, meas: list[F]) -> list[F]:
return meas[:self.length]
def decode(
self,
output: list[F],
_num_measurements: int) -> list[int]:
return [bucket_count.int() for
bucket_count in output]
8.
Poplar1
This section specifies Poplar1, a VDAF for the following task. Each Client
holds a bit-string of length
BITS
and the Collector chooses a sequence of
-bit strings, where
L <= BITS
. The latter is referred to as the
"candidate prefixes". The goal is to count how many of the Clients' inputs
begin with each candidate prefix.
This functionality is the core component of the heavy hitters protocol of
BBCGGI21
. The goal of this protocol is to compute the subset of inputs held
by at least
Clients for some threshold
. It invokes Poplar1 as follows:
Each Client shards its string into secret shares and uploads one share to
each of the Aggregators.
The Collector picks an initial set of candidate prefixes, say
and
and sends them to the Aggregators.
The Aggregators run Poplar1 verification and aggregation on each of the
reports and send their aggregate shares to the Collector.
The Collector unshards the aggregate result, which consists of the hit count
for each candidate prefix. For each prefix
with hit count at least
the Collector adds
p || 0
and
p || 1
to the next generation of
candidate prefixes and repeats Step 2.
While Poplar1 is intended to be used to compute heavy hitters in the above
protocol, it may be possible to use it for other applications as well. However,
care must be taken to ensure such usage is secure. See
Section 9.4
and
Section 9.5
for details.
Poplar1 is constructed from an "Incremental Distributed Point Function (IDPF)",
a primitive described by
BBCGGI21
that generalizes the notion of a
Distributed Point Function (DPF)
GI14
. Briefly, a DPF is used to distribute
the computation of a "point function", a function that evaluates to zero on
every input except at a programmable "point". The computation is distributed in
such a way that no one party knows either the point or what it evaluates to.
An IDPF generalizes this "point" to a path on a full binary tree from the root
to one of the leaves. It is evaluated on an "index" representing a unique node
of the tree. If the node is on the programmed path, then the function evaluates
to a non-zero value; otherwise it evaluates to zero. This structure allows an
IDPF to provide the functionality required for the above protocol: to compute
the hit count for an index, just evaluate each set of IDPF shares at that index
and add up the results.
Consider the sub-tree constructed from a set of input strings and a target
threshold
by including all indices with hit count at least
. This
structure is called the "prefix tree" of the batch of measurements and target
threshold. To compute the
-heavy-hitters for the batch, the Aggregators and
Collector first compute the prefix tree, then extract the heavy hitters from
the leaves of this tree. Note that the prefix tree leaks more information about
the set than the heavy hitters themselves; see
Section 9.4
for more
discussion.
Poplar1 composes an IDPF with the arithmetic sketch of
BBCGGI21
, Section
4.2. (The paper calls this a "secure sketch", but the underlying technique was
later generalized in
BBCGGI23
, where it is called "arithmetic sketching".)
The sketch ensures that evaluating a set of input shares on a set of unique
candidate prefixes results in shares of a zero vector or a "one-hot" vector,
i.e., a vector that is zero everywhere except for in at most one position.
Moreover, the value at that position should be one.
The remainder of this section is structured as follows. The syntax of IDPFs is
defined in
Section 8.1
. The Poplar1 VDAF is defined in
Section 8.2
in
terms of a generic IDPF. A specification of the IDPF of
BBCGGI21
is given in
Section 8.3
. Test vectors for Poplar1 can be found in
Appendix C
8.1.
Incremental Distributed Point Functions (IDPFs)
An IDPF is defined over a domain of size
2**BITS
, where
BITS
is a constant.
Indices into the IDPF tree are bit strings. (In Poplar1, each Client's bit
string is an index; see
Section 8.1.1
for details.) The Client
specifies an index
alpha
and a vector of values
beta
, one for each "level"
in the range
[0, BITS)
. The key generation algorithm generates one IDPF
"key" for each Aggregator. When evaluated at level
and index
prefix
, each
IDPF key returns an additive share of
beta[L]
if
prefix
is the
-bit
prefix of
alpha
and shares of zero otherwise.
Each of the programmed points
beta
is a vector of elements of some finite
field. There are two types of fields: one for inner nodes (denoted
FieldInner
), and one for leaf nodes (
FieldLeaf
). (The instantiation of
Poplar1 (
Section 8.2
) will use a much larger field for leaf nodes
than for inner nodes. This is to ensure the IDPF is "extractable" as defined in
BBCGGI21
, Definition 1. See
Section 9.5
for details.)
A concrete IDPF defines the types and parameters enumerated in
Table 15
In the remainder,
Output
is used as shorthand for the type
list[list[FieldInner]] | list[list[FieldLeaf]]
. (This type denotes either a
vector of inner node field elements or leaf node field elements.) The scheme is
comprised of the following algorithms:
idpf.gen(alpha: tuple[bool, ...], beta_inner: list[list[FieldInner]], beta_leaf:
list[FieldLeaf], ctx: bytes, nonce: bytes, rand: bytes) -> tuple[PublicShare,
list[bytes]]
is the IDPF-key generation algorithm. Its inputs are the index
alpha
, the values
beta
, the application context, and the report nonce.
The output is a public part (of type
PublicShare
) that is sent to each
Aggregator and a vector of private IDPF keys, one for each Aggregator. The
nonce and application context are used to derive the fixed AES key for
XofFixedKeyAes128 (
Section 6.2.2
). Looking ahead, this key is used
for extending a node's seed into the seeds for the child nodes at each level
of the tree; see
Section 8.3
Pre-conditions:
alpha
MUST
have length
BITS
beta_inner
MUST
have length
BITS - 1
beta_inner[level]
MUST
have length
VALUE_LEN
for each
level
in
the range
[0, BITS - 1)
beta_leaf
MUST
have length
VALUE_LEN
rand
MUST
be generated by a CSPRNG and have length
RAND_SIZE
nonce
MUST
be generated by a CSPRNG (see
Section 9.2
for
details) and have length
idpf.NONCE_SIZE
Post-conditions:
The number of IDPF keys
MUST
be
idpf.SHARES
idpf.eval(agg_id: int, public_share: PublicShare, key: bytes, level: int,
prefixes: Sequence[tuple[bool, ...]], ctx: bytes, nonce: bytes) -> Output
is
the IDPF-key evaluation algorithm run by each Aggregator. Its inputs are the
Aggregator's unique identifier, the public share distributed to all of the
Aggregators, the Aggregator's IDPF key, the "level" at which to evaluate the
IDPF, the sequence of candidate prefixes, the application context, and the
report nonce. It returns the share of the value corresponding to each
candidate prefix.
Pre-conditions:
agg_id
MUST
be in the range
[0, idpf.SHARES)
and match the index of
key
in the sequence of IDPF keys output by the Client.
level
MUST
be in the range
[0, BITS)
Each prefix
MUST
be distinct and have length
level + 1
The length of the nonce
MUST
be
idpf.NONCE_SIZE
Post-conditions:
The length of the output
MUST
be
len(prefixes)
The length of each element of the output
MUST
be
idpf.VALUE_LEN
If
level == idpf.BITS - 1
, then the output field
MUST
be
FieldLeaf
and
FieldInner
otherwise
In addition, the IDPF provides the following method:
def current_field(
self,
level: int) -> type[FieldInner] | type[FieldLeaf]:
if level < self.BITS - 1:
return self.field_inner
return self.field_leaf
Finally, an implementation note. The interface for IDPFs specified here is
stateless, in the sense that there is no state carried between IDPF evaluations.
This is to align the IDPF syntax with the VDAF abstraction boundary, which does
not include shared state across VDAF evaluations. In practice, of course, it
will often be beneficial to expose a stateful API for IDPFs and carry the state
across evaluations. See
Section 8.3
for details.
Table 15
Constants and types defined by a concrete IDPF.
Parameter
Description
SHARES: int
Number of IDPF keys output by IDPF-key generator.
BITS: int
Length in bits of each input string.
VALUE_LEN: int
Number of field elements of each output value.
RAND_SIZE: int
Size of the random string consumed by the IDPF-key generator.
NONCE_SIZE: int
Size of the random nonce generated by the Client.
KEY_SIZE: int
Size in bytes of each IDPF key.
FieldInner
Implementation of
Field
Section 6.1
) used for values of inner nodes.
FieldLeaf
Implementation of
Field
used for values of leaf nodes.
PublicShare
Type of public share for this IDPF.
Output
Alias of
list[list[FieldInner]] | list[list[FieldLeaf]]
FieldVec
Alias of
list[FieldInner] | list[FieldLeaf].
8.1.1.
Encoding Inputs as Indices
How data are represented as IDPF indices is up to the application. When the
inputs are fixed-length byte strings, the most natural choice of representation
is as a bit string formed from all the bits of the byte string, first ordered by
byte position, then ordered from most significant bit to least significant bit
within each byte. This ensures that, when a byte string is a prefix of another,
so too is its corresponding index. (Index prefixes are defined in
Section 8.1
.) For
example,
Byte string: 01 02
Bit string: 00000001 00000010
is a prefix of
Byte string: 01 02 03
Bit string: 00000001 00000010 00000011
Additionally, lexicographic ordering is preserved by this mapping from a byte
string to a bit string.
When the inputs are variable length, it is necessary to pad each input to some
fixed length. Further, the padding scheme must be non-ambiguous. For example,
each input could be padded with
b"\x01"
followed by as many
b"\x00"
bytes
as needed.
8.2.
Specification
This section specifies
Poplar1
, an implementation of the
Vdaf
interface
Section 5
). It is defined in terms of the
Idpf
implementation of
Section 8.3
with
SHARES == 2
and
VALUE_LEN == 2
and
XofTurboShake128
as specified in
Section 6.2.1
. The associated
constants and types required by the
Vdaf
interface are defined in
Table 16
. The methods required for sharding, verification,
aggregation, and unsharding are described in the remaining subsections. These
methods make use of constants defined in
Table 17
Table 16
VDAF parameters for Poplar1.
Parameter
Value
idpf
As specified in
Section 8.3
xof
XofTurboShake128
Section 6.2.1
VERIFY_KEY_SIZE
xof.SEED_SIZE
RAND_SIZE
xof.SEED_SIZE * 3 + idpf.RAND_SIZE
NONCE_SIZE
16
ROUNDS
SHARES
Measurement
tuple[bool, ...]
AggParam
tuple[int, Sequence[tuple[bool, ...]]]
PublicShare
As defined by
idpf
InputShare
tuple[bytes, bytes, list[FieldInner], list[FieldLeaf]]
OutShare
FieldVec
AggShare
FieldVec
AggResult
list[int]
VerifyState
tuple[bytes, int, FieldVec]
VerifierShare
FieldVec
VerifierMessage
Optional[FieldVec]
Table 17
Constants used by Poplar1.
Variable
Value
USAGE_SHARD_RAND: int
USAGE_CORR_INNER: int
USAGE_CORR_LEAF: int
USAGE_VERIFY_RAND: int
8.2.1.
Sharding
The Client's measurement is an IDPF index, denoted
alpha
, whose type is a
sequence of bits
tuple[bool, ...]
(See
Section 8.1.1
for
guidelines on index encoding.)
The programmed IDPF values are pairs of field elements
(1, k)
where each
is
chosen at random. This random value is used as part of the arithmetic sketching
protocol of
BBCGGI21
, Appendix C.4. After evaluating their IDPF key shares
on a given sequence of candidate prefixes, the Aggregators use the sketching
protocol to verify that they hold shares of a zero vector or a one-hot vector at
a given level of the IDPF tree.
In addition to programming
into the IDPF output, for each level of the
tree, the Client generates random elements
, and
and computes
A = -2*a + k
B = a**2 + b - k*a + c
and sends additive shares of
and
to each of the
Aggregators. These help the Aggregators evaluate the sketch during
verification.
Putting everything together, the sharding algorithm is defined as
follows.
def shard(
self,
ctx: bytes,
measurement: tuple[bool, ...],
nonce: bytes,
rand: bytes,
) -> tuple[Poplar1PublicShare, list[Poplar1InputShare]]:
if len(nonce) != self.NONCE_SIZE:
raise ValueError("incorrect nonce size")
if len(rand) != self.RAND_SIZE:
raise ValueError("incorrect size of random bytes argument")
l = self.xof.SEED_SIZE
# Split the random input into the random input for IDPF key
# generation, correlated randomness, and sharding.
if len(rand) != self.RAND_SIZE:
raise ValueError('incorrect rand size')
idpf_rand, rand = front(self.idpf.RAND_SIZE, rand)
seeds = [rand[i:i + l] for i in range(0, 3 * l, l)]
corr_seed, seeds = front(2, seeds)
(shard_seed,), seeds = front(1, seeds)
xof = self.xof(
shard_seed,
self.domain_separation_tag(USAGE_SHARD_RAND, ctx),
nonce,
# Construct the IDPF values for each level of the IDPF tree.
# Each "data" value is 1; in addition, the Client generates
# a random "authenticator" value used by the Aggregators to
# evaluate the sketch during verification. This sketch is
# used to verify the one-hotness of their output shares.
beta_inner = [
[self.idpf.field_inner(1), k]
for k in xof.next_vec(self.idpf.field_inner,
self.idpf.BITS - 1)
beta_leaf = [self.idpf.field_leaf(1)] + \
xof.next_vec(self.idpf.field_leaf, 1)
# Generate the IDPF keys.
(public_share, keys) = self.idpf.gen(
measurement,
beta_inner,
beta_leaf,
ctx,
nonce,
idpf_rand,
# Generate correlated randomness used by the Aggregators to
# evaluate the sketch over their output shares. Seeds are used
# to encode shares of the `(a, b, c)` triples. (See [BBCGGI21,
# Appendix C.4].)
corr_offsets: list[Field] = vec_add(
self.xof.expand_into_vec(
self.idpf.field_inner,
corr_seed[0],
self.domain_separation_tag(USAGE_CORR_INNER, ctx),
byte(0) + nonce,
3 * (self.idpf.BITS - 1),
),
self.xof.expand_into_vec(
self.idpf.field_inner,
corr_seed[1],
self.domain_separation_tag(USAGE_CORR_INNER, ctx),
byte(1) + nonce,
3 * (self.idpf.BITS - 1),
),
corr_offsets += vec_add(
self.xof.expand_into_vec(
self.idpf.field_leaf,
corr_seed[0],
self.domain_separation_tag(USAGE_CORR_LEAF, ctx),
byte(0) + nonce,
3,
),
self.xof.expand_into_vec(
self.idpf.field_leaf,
corr_seed[1],
self.domain_separation_tag(USAGE_CORR_LEAF, ctx),
byte(1) + nonce,
3,
),
# For each level of the IDPF tree, shares of the `(A, B)`
# pairs are computed from the corresponding `(a, b, c)`
# triple and authenticator value `k`.
corr_inner: list[list[Field64]] = [[], []]
for level in range(self.idpf.BITS):
field = cast(type[Field], self.idpf.current_field(level))
k = beta_inner[level][1] if level < self.idpf.BITS - 1 \
else beta_leaf[1]
(a, b, c), corr_offsets = corr_offsets[:3], corr_offsets[3:]
A = -field(2) * a + k
B = a ** 2 + b - a * k + c
corr1 = xof.next_vec(field, 2)
corr0 = vec_sub([A, B], corr1)
if level < self.idpf.BITS - 1:
corr_inner[0] += cast(list[Field64], corr0)
corr_inner[1] += cast(list[Field64], corr1)
else:
corr_leaf = [
cast(list[Field255], corr0),
cast(list[Field255], corr1),
# Each input share consists of the Aggregator's IDPF key
# and a share of the correlated randomness.
input_shares = list(zip(keys, corr_seed, corr_inner, corr_leaf))
return (public_share, input_shares)
8.2.2.
Verification
The aggregation parameter encodes a sequence of candidate prefixes. When an
Aggregator receives an input share from the Client, it begins by evaluating its
IDPF share on each candidate prefix, recovering a
data_share
and
auth_share
for each. The Aggregators use these and the correlation shares provided by the
Client to verify that the sequence of
data_share
values are additive shares of
a zero vector or a one-hot vector.
Aggregators
MUST
ensure the candidate prefixes are all unique and appear in
lexicographic order. (This is enforced in the definition of
is_valid()
below.) Uniqueness is necessary to ensure the refined measurement (i.e., the sum
of the output shares) is in fact a one-hot vector. Otherwise, sketch
verification might fail, causing the Aggregators to erroneously reject a report
that is actually valid. Note that enforcing the order is not strictly necessary,
but this does allow uniqueness to be determined more efficiently.
def verify_init(
self,
verify_key: bytes,
ctx: bytes,
agg_id: int,
agg_param: Poplar1AggParam,
nonce: bytes,
public_share: Poplar1PublicShare,
input_share: Poplar1InputShare) -> tuple[
Poplar1VerifyState,
FieldVec]:
(level, prefixes) = agg_param
(key, corr_seed, corr_inner, corr_leaf) = input_share
field = self.idpf.current_field(level)
# Evaluate the IDPF key at the given set of prefixes.
value = self.idpf.eval(
agg_id, public_share, key, level, prefixes, ctx, nonce)
# Get shares of the correlated randomness for evaluating the
# Aggregator's share of the sketch.
if level < self.idpf.BITS - 1:
corr_xof = self.xof(
corr_seed,
self.domain_separation_tag(USAGE_CORR_INNER, ctx),
byte(agg_id) + nonce,
# Fast-forward the XOF state to the current level.
corr_xof.next_vec(field, 3 * level)
else:
corr_xof = self.xof(
corr_seed,
self.domain_separation_tag(USAGE_CORR_LEAF, ctx),
byte(agg_id) + nonce,
(a_share, b_share, c_share) = corr_xof.next_vec(field, 3)
if level < self.idpf.BITS - 1:
(A_share, B_share) = cast(
list[Field],
corr_inner[2 * level:2 * (level + 1)],
else:
(A_share, B_share) = cast(list[Field], corr_leaf)
# Evaluate the Aggregator's share of the sketch. These are
# called the "masked input values" [BBCGGI21, Appendix C.4].
verify_rand_xof = self.xof(
verify_key,
self.domain_separation_tag(USAGE_VERIFY_RAND, ctx),
nonce + to_be_bytes(level, 2),
verify_rand = cast(
list[Field],
verify_rand_xof.next_vec(field, len(prefixes)),
sketch_share = cast(
list[Field],
[a_share, b_share, c_share],
out_share = []
for (i, r) in enumerate(verify_rand):
data_share = cast(Field, value[i][0])
auth_share = cast(Field, value[i][1])
sketch_share[0] += data_share * r
sketch_share[1] += data_share * r ** 2
sketch_share[2] += auth_share * r
out_share.append(data_share)
verify_mem = [A_share, B_share, field(agg_id)] + out_share
return (
b'evaluate sketch',
level,
cast(FieldVec, verify_mem),
),
cast(FieldVec, sketch_share),
def verify_next(
self,
_ctx: bytes,
verify_state: Poplar1VerifyState,
verifier_message: Optional[FieldVec]
) -> tuple[Poplar1VerifyState, FieldVec] | FieldVec:
prev_sketch = cast(list[Field], verifier_message)
(step, level, verify_mem) = verify_state
if step == b'evaluate sketch':
if prev_sketch is None:
raise ValueError('expected value, got none')
elif len(prev_sketch) != 3:
raise ValueError('incorrect sketch length')
A_share = cast(Field, verify_mem[0])
B_share = cast(Field, verify_mem[1])
agg_id = cast(Field, verify_mem[2])
verify_mem = verify_mem[3:]
sketch_share = [
agg_id * (prev_sketch[0] ** 2
- prev_sketch[1]
- prev_sketch[2])
+ A_share * prev_sketch[0]
+ B_share
return cast(
tuple[Poplar1VerifyState, FieldVec],
b'reveal sketch',
level,
verify_mem,
),
sketch_share,
elif step == b'reveal sketch':
if prev_sketch is None:
return verify_mem # Output shares
else:
raise ValueError('invalid verifier message')
raise ValueError('invalid verification state')
def verifier_shares_to_message(
self,
_ctx: bytes,
agg_param: Poplar1AggParam,
verifier_shares: list[FieldVec]) -> Optional[FieldVec]:
if len(verifier_shares) != 2:
raise ValueError('incorrect number of verifier shares')
(level, _) = agg_param
field = self.idpf.current_field(level)
sketch = vec_add(
cast(list[Field], verifier_shares[0]),
cast(list[Field], verifier_shares[1]),
if len(sketch) == 3:
return cast(FieldVec, sketch)
elif len(sketch) == 1:
if sketch == field.zeros(1):
# In order to reduce communication overhead, let `None`
# denote a successful sketch verification.
return None
else:
raise ValueError('sketch verification failed')
else:
raise ValueError('incorrect sketch length')
8.2.3.
Validity of Aggregation Parameters
Aggregation parameters are valid for a given input share if no aggregation
parameter with the same level has been used with the same input share before.
The verification phase
MUST NOT
be run more than once for a given combination
of input share and level. This function checks that candidate prefixes are
unique and lexicographically sorted, checks that levels are increasing between
calls, and also enforces that the prefixes at each level are suffixes of the
previous level's prefixes.
def is_valid(
self,
agg_param: Poplar1AggParam,
previous_agg_params: list[Poplar1AggParam]) -> bool:
"""
Checks that candidate prefixes are unique and lexicographically
sorted, checks that levels are increasing between calls, and also
enforces that the prefixes at each level are suffixes of the
previous level's prefixes.
"""
(level, prefixes) = agg_param
# Ensure that candidate prefixes are all unique and appear in
# lexicographic order.
for i in range(1, len(prefixes)):
if prefixes[i - 1] >= prefixes[i]:
return False
if len(previous_agg_params) < 1:
return True
(last_level, last_prefixes) = previous_agg_params[-1]
last_prefixes_set = set(last_prefixes)
# Check that level increased.
if level <= last_level:
return False
# Check that prefixes are suffixes of the last level's prefixes.
for prefix in prefixes:
last_prefix = get_ancestor(prefix, last_level)
if last_prefix not in last_prefixes_set:
# Current prefix not a suffix of last level's prefixes.
return False
return True
def get_ancestor(
index: tuple[bool, ...],
level: int) -> tuple[bool, ...]:
"""
Helper function to determine the prefix of `index` at
`level`.
"""
return index[:level + 1]
8.2.4.
Aggregation
Aggregation involves simply adding up the output shares.
def agg_init(self, agg_param: Poplar1AggParam) -> FieldVec:
(level, prefixes) = agg_param
field = self.idpf.current_field(level)
return field.zeros(len(prefixes))
def agg_update(self,
agg_param: Poplar1AggParam,
agg_share: FieldVec,
out_share: FieldVec) -> FieldVec:
a = cast(list[Field], agg_share)
o = cast(list[Field], out_share)
return cast(FieldVec, vec_add(a, o))
def merge(self,
agg_param: Poplar1AggParam,
agg_shares: list[FieldVec]) -> FieldVec:
(level, prefixes) = agg_param
field = self.idpf.current_field(level)
agg = cast(list[Field], field.zeros(len(prefixes)))
for agg_share in agg_shares:
agg = vec_add(agg, cast(list[Field], agg_share))
return cast(FieldVec, agg)
8.2.5.
Unsharding
Finally, the Collector unshards the aggregate result by adding up the aggregate
shares.
def unshard(
self,
agg_param: Poplar1AggParam,
agg_shares: list[FieldVec],
_num_measurements: int) -> list[int]:
agg = self.merge(agg_param, agg_shares)
return [x.int() for x in agg]
8.2.6.
Message Serialization
This section defines serialization formats for messages exchanged over the
network while executing
Poplar1
. Messages are defined in the presentation
language of TLS as defined in
Section 3
of [
RFC8446
Let
poplar1
be an instance of
Poplar1
. In the remainder let
Fi
be an
alias for
poplar1.idpf.field_inner.ENCODED_SIZE
Fl
as an alias for
poplar1.idpf.field_leaf.ENCODED_SIZE
, and
as an alias for
poplar1.idpf.BITS
Elements of the inner field are encoded in little-endian byte order (as defined
in
Section 6.1
) and are represented as follows:
opaque Poplar1FieldInner[Fi];
Likewise, elements of the leaf field are encoded in little-endian byte order
(as defined in
Section 6.1
) and are represented as follows:
opaque Poplar1FieldLeaf[Fl];
8.2.6.1.
Public Share
The public share of the IDPF scheme in
Section 8.3
consists of a sequence
of "correction words". A correction word has three components:
the XOF seed of type
bytes
the control bits of type
tuple[bool, bool]
; and
the payload of type
list[Field64]
for the first
BITS-1
words and
list[Field255]
for the last word.
The encoding is a straightforward structure of arrays, except that the control
bits are packed as tightly as possible. The encoded public share is structured
as follows:
struct {
opaque packed_control_bits[packed_len];
opaque seed[poplar1.idpf.KEY_SIZE*B];
Poplar1FieldInner payload_inner[Fi*poplar1.idpf.VALUE_LEN*(B-1)];
Poplar1FieldLeaf payload_leaf[Fl*poplar1.idpf.VALUE_LEN];
} Poplar1PublicShare;
Here
packed_len = (2*B + 7) // 8
is the length of the packed control bits.
Field
packed_control_bits
is encoded with the following function:
packed_control_buf = [int(0)] * packed_len
for i, bit in enumerate(control_bits):
packed_control_buf[i // 8] |= bit << (i % 8)
packed_control_bits = bytes(packed_control_buf)
It encodes each group of eight bits into a byte, in LSB to MSB order, padding the most
significant bits of the last byte with zeros as necessary, and returns the byte
array. Decoding performs the reverse operation: it takes in a byte array
and a number of bits, and returns a list of bits, extracting eight bits from
each byte in turn, in LSB to MSB order, and stopping after the requested number
of bits. If the byte array has an incorrect length, or if unused bits in the
last bytes are not zero, it throws an error:
control_bits = []
for i in range(length):
control_bits.append(bool(
(packed_control_bits[i // 8] >> (i % 8)) & 1
))
leftover_bits = packed_control_bits[-1] >> (
(length + 7) % 8 + 1
if (length + 7) // 8 != len(packed_control_bits) or \
leftover_bits != 0:
raise ValueError('trailing bits')
8.2.6.2.
Input Share
Each input share is structured as follows:
struct {
opaque idpf_key[poplar1.idpf.KEY_SIZE];
opaque corr_seed[poplar1.xof.SEED_SIZE];
Poplar1FieldInner corr_inner[Fi * 2 * (B- 1)];
Poplar1FieldLeaf corr_leaf[Fl * 2];
} Poplar1InputShare;
8.2.6.3.
Verifier Share
Encoding of the verifier share depends on the round of sketching: if the first
round, then each sketch share has three field elements; if the second round,
then each sketch share has one field element. The field that is used depends on
the level of the IDPF tree specified by the aggregation parameter, either the
inner field or the leaf field.
For the first round and inner field:
struct {
Poplar1FieldInner sketch_share[Fi * 3];
} Poplar1VerifierShareRoundOneInner;
For the first round and leaf field:
struct {
Poplar1FieldLeaf sketch_share[Fl * 3];
} Poplar1VerifierShareRoundOneLeaf;
For the second round and inner field:
struct {
Poplar1FieldInner sketch_share;
} Poplar1VerifierShareRoundTwoInner;
For the second round and leaf field:
struct {
Poplar1FieldLeaf sketch_share;
} Poplar1VerifierShareRoundTwoLeaf;
8.2.6.4.
Verifier Message
Likewise, the structure of the verifier message for Poplar1 depends on the
sketching round and field. For the first round and inner field:
struct {
Poplar1FieldInner[Fi * 3];
} Poplar1VerifierMessageRoundOneInner;
For the first round and leaf field:
struct {
Poplar1FieldLeaf sketch[Fl * 3];
} Poplar1VerifierMessageRoundOneLeaf;
Note that these messages have the same structures as the verifier shares for
the first round.
The second-round verifier message is the empty string. This is because the
sketch shares are expected to sum to a particular value if the output shares
are valid; successful verification is represented with the empty string,
otherwise the procedure returns an error.
8.2.6.5.
Aggregate Share
The encoding of the aggregate share depends on whether the inner or leaf field
is used, and the number of candidate prefixes. Both of these are determined by
the aggregation parameter.
Let
prefix_count
denote the number of candidate prefixes. For the inner field:
struct {
Poplar1FieldInner agg_share[Fi * prefix_count];
} Poplar1AggShareInner;
For the leaf field:
struct {
Poplar1FieldLeaf agg_share[Fl * prefix_count];
} Poplar1AggShareLeaf;
8.2.6.6.
Aggregation Parameter
The aggregation parameter is encoded as follows:
struct {
uint16_t level;
uint32_t num_prefixes;
opaque encoded_prefixes[prefixes_len];
} Poplar1AggParam;
The fields in this struct are:
level
, the level of the IDPF tree of each
prefixes;
num_prefixes
, the number of prefixes to evaluate; and
encoded_prefixes
, the sequence of prefixes encoded into a byte string of
length
prefixes_len
. Each prefix is packed into a byte string, with the bits
assigned in MSB-to-LSB order, and then the byte strings for each prefix are
concatenated together. The prefixes are encoded with the following procedure:
prefixes_len = ((level + 1) + 7) // 8 * len(prefixes)
encoded_prefixes = bytearray()
for prefix in prefixes:
for chunk in itertools.batched(prefix, 8):
byte_out = 0
for (bit_position, bit) in enumerate(chunk):
byte_out |= bit << (7 - bit_position)
encoded_prefixes.append(byte_out)
Decoding involves the following procedure:
prefixes = []
last_byte_mask = 0
leftover_bits = (level + 1) % 8
if leftover_bits > 0:
for bit_index in range(8 - leftover_bits, 8):
last_byte_mask |= 1 << bit_index
last_byte_mask ^= 255
bytes_per_prefix = ((level + 1) + 7) // 8
for chunk in itertools.batched(encoded_prefixes, bytes_per_prefix):
if chunk[-1] & last_byte_mask > 0:
raise ValueError('trailing bits in prefix')
prefix = []
for i in range(level + 1):
byte_index = i // 8
bit_offset = 7 - (i % 8)
bit = (chunk[byte_index] >> bit_offset) & 1 != 0
prefix.append(bit)
prefixes.append(tuple(prefix))
Implementation note: the aggregation parameter includes the level of the IDPF
tree and the sequence of indices to evaluate. For implementations that perform
per-report caching across executions of the VDAF, this may be more information
than is strictly needed. In particular, it may be sufficient to convey which
indices from the previous execution will have their children included in the
next. This would help reduce communication overhead.
8.3.
IDPF Specification
This section specifies a concrete IDPF suitable for instantiating Poplar1. The
constant and type definitions required by the
Idpf
interface are given in
Table 18
The IDPF requires an XOF for deriving the output shares, as well as a variety
of other artifacts used internally. For performance reasons, this object is
instantiated using XofFixedKeyAes128 (
Section 6.2.2
) wherever
possible. See
Section 9.6
for security considerations.
Table 18
Constants and type definitions for the concrete IDPF.
Parameter
Value
xof
XofFixedKeyAes128
Section 6.2.2
SHARES
BITS
Any positive integer.
VALUE_LEN
Any positive integer.
KEY_SIZE
xof.SEED_SIZE
FieldInner
Field64
Table 4
FieldLeaf
Field255
Table 4
8.3.1.
Overview
At a high level, the IDPF maps a key generator's input
(alpha, beta_1, ...,
beta_BITS)
onto a binary tree with
BITS+1
levels, where each edge going from
a parent node to a left child is labeled
, and each right edge is labeled
. Then each leaf node corresponds to a bit string of length
BITS
, where the
labels on the path from the root to
contain the individual bits. Finally,
all nodes in the tree have an assigned value, with the nodes on the path
from the root to
alpha
having values
beta_1, ..., beta_BITS
, and all other
nodes having value
The IDPF construction now boils down to secret-sharing the values at each node
of that tree in an efficient way. Note that explicitly representing the tree
requires
O(2**BITS)
space, so the generator cannot just compute additive
shares of it and send them to the two evaluators. Instead, the evaluators will
re-generate shares of values at selected nodes of the tree using a XOF
Section 6.2
).
The basic observation is that if both evaluators have the same seed
of
length
KEY_SIZE
, then expanding
using a XOF will also result in the same
expansion. If the length of the XOF expansion is set to
2*KEY_SIZE
, it can
then be split again into two seeds
s_l
s_r
, that can again serve as XOF
seeds. Now, viewing the seeds as XOR-shares of integers, if evaluators have the
same seed at the root of the tree, then their expanded trees will form a
secret-shared tree of zeros. The actual construction will additionally use a
convert()
function before each expansion, which maps seeds into the
appropriate output domain (see
Section 8.3.4
), generating
a new seed for the next level in the process.
The open task now is to ensure that evaluators have different seeds at nodes
that lie on the path to
alpha
, while having the same seeds on all other nodes.
This is done using so-called "correction words" included in the public share.
The correction words are conditionally added to the XOF output by both
evaluators. The condition here is a secret-shared bit, called a "control bit",
which indicates whether the current node is on the path to
alpha
or not. On
the path, the control bits add up to
, meaning only one evaluator will add
the correction word to its XOF output. Off the path, either none or both
evaluators add the correction word, and so the seeds at the next level stay the
same.
What remains is to turn the (now pseudorandom) values on the path to
alpha
into the desired
beta
values. This is done by including "value correction
words" in the public share, which are chosen such that when added with the
pseudorandom shares at the
th node on the path to
alpha
, they add up to
shares of
beta_i
The following two sections describe the algorithms for key generation in full
detail.
8.3.2.
Key Generation
The description of the IDPF-key generation algorithm makes use of auxiliary
functions
extend()
and
convert()
defined in
Section 8.3.4
def gen(
self,
alpha: tuple[bool, ...],
beta_inner: list[list[Field64]],
beta_leaf: list[Field255],
ctx: bytes,
nonce: bytes,
rand: bytes) -> tuple[list[CorrectionWord], list[bytes]]:
if len(alpha) != self.BITS:
raise ValueError("incorrect alpha length")
if len(beta_inner) != self.BITS - 1:
raise ValueError("incorrect beta_inner length")
if len(rand) != self.RAND_SIZE:
raise ValueError("incorrect rand size")
if len(nonce) != self.NONCE_SIZE:
raise ValueError("incorrect nonce size")
key = [
rand[:XofFixedKeyAes128.SEED_SIZE],
rand[XofFixedKeyAes128.SEED_SIZE:],
seed = key.copy()
ctrl = [False, True]
public_share = []
for level in range(self.BITS):
bit = alpha[level]
keep = int(bit)
lose = 1 - keep
(s0, t0) = self.extend(level, seed[0], ctx, nonce)
(s1, t1) = self.extend(level, seed[1], ctx, nonce)
seed_cw = xor(s0[lose], s1[lose])
ctrl_cw = (
t0[0] ^ t1[0] ^ (not bit),
t0[1] ^ t1[1] ^ bit,
# Implementation note: these conditional XORs and
# input-dependent array indices should be replaced with
# constant-time selects in practice in order to reduce
# leakage via timing side channels.
if ctrl[0]:
x0 = xor(s0[keep], seed_cw)
ctrl[0] = t0[keep] ^ ctrl_cw[keep]
else:
x0 = s0[keep]
ctrl[0] = t0[keep]
if ctrl[1]:
x1 = xor(s1[keep], seed_cw)
ctrl[1] = t1[keep] ^ ctrl_cw[keep]
else:
x1 = s1[keep]
ctrl[1] = t1[keep]
(seed[0], w0) = self.convert(level, x0, ctx, nonce)
(seed[1], w1) = self.convert(level, x1, ctx, nonce)
if level < self.BITS - 1:
b = cast(list[Field], beta_inner[level])
else:
b = cast(list[Field], beta_leaf)
if len(b) != self.VALUE_LEN:
raise ValueError(
"length of beta must match the value length"
w_cw = vec_add(vec_sub(b, w0), w1)
# Implementation note: this conditional negation should be
# replaced with a constant time select or a constant time
# multiplication in practice in order to reduce leakage via
# timing side channels.
if ctrl[1]:
for i in range(len(w_cw)):
w_cw[i] = -w_cw[i]
public_share.append((seed_cw, ctrl_cw, w_cw))
return (public_share, key)
8.3.3.
Key Evaluation
The description of the IDPF-evaluation algorithm makes use of auxiliary
functions
extend()
and
convert()
defined in
Section 8.3.4
def eval(
self,
agg_id: int,
public_share: list[CorrectionWord],
key: bytes,
level: int,
prefixes: Sequence[tuple[bool, ...]],
ctx: bytes,
nonce: bytes) -> list[list[Field64]] | list[list[Field255]]:
if agg_id not in range(self.SHARES):
raise ValueError('aggregator id out of range')
if level not in range(self.BITS):
raise ValueError('level out of range')
if len(set(prefixes)) != len(prefixes):
raise ValueError('prefixes must be unique')
out_share = []
for prefix in prefixes:
if len(prefix) != level + 1:
raise ValueError('incorrect prefix length')
# The Aggregator's output share is the value of a node of
# the IDPF tree at the given `level`. The node's value is
# computed by traversing the path defined by the candidate
# `prefix`. Each node in the tree is represented by a seed
# (`seed`) and a control bit (`ctrl`).
seed = key
ctrl = bool(agg_id)
y: FieldVec
for current_level in range(level + 1):
bit = int(prefix[current_level])
# Implementation note: typically the current round of
# candidate prefixes would have been derived from
# aggregate results computed during previous rounds.
# For example, when using the IDPF to compute heavy
# hitters, a string whose hit count exceeded the
# given threshold in the last round would be the
# prefix of each `prefix` in the current round. (See
# [BBCGGI21, Section 5.1].) In this case, part of the
# path would have already been traversed.
# Re-computing nodes along previously traversed paths is
# wasteful. Implementations can eliminate this added
# complexity by caching nodes (i.e., `(seed, ctrl)`
# pairs) output by previous calls to `eval_next()`.
(seed, ctrl, y) = self.eval_next(
seed,
ctrl,
public_share[current_level],
current_level,
bit,
ctx,
nonce,
if agg_id == 0:
out_share.append(cast(list[Field], y))
else:
out_share.append(vec_neg(cast(list[Field], y)))
return cast(
list[list[Field64]] | list[list[Field255]],
out_share,
def eval_next(
self,
prev_seed: bytes,
prev_ctrl: bool,
correction_word: CorrectionWord,
level: int,
bit: int,
ctx: bytes,
nonce: bytes) -> tuple[bytes, bool, FieldVec]:
"""
Compute the next node in the IDPF tree along the path determined
by a candidate prefix. The next node is determined by `bit`, the
bit of the prefix corresponding to the next level of the tree.
"""
seed_cw = correction_word[0]
ctrl_cw = correction_word[1]
w_cw = cast(list[Field], correction_word[2])
(s, t) = self.extend(level, prev_seed, ctx, nonce)
# Implementation note: these conditional operations and
# input-dependent array indices should be replaced with
# constant-time selects in practice in order to reduce leakage
# via timing side channels.
if prev_ctrl:
s[0] = xor(s[0], seed_cw)
s[1] = xor(s[1], seed_cw)
t[0] ^= ctrl_cw[0]
t[1] ^= ctrl_cw[1]
next_ctrl = t[bit]
convert_output = self.convert(level, s[bit], ctx, nonce)
next_seed = convert_output[0]
y = cast(list[Field], convert_output[1])
# Implementation note: this conditional addition should be
# replaced with a constant-time select in practice in order to
# reduce leakage via timing side channels.
if next_ctrl:
for i in range(len(y)):
y[i] += w_cw[i]
return (next_seed, next_ctrl, cast(FieldVec, y))
8.3.4.
Auxiliary Functions
def extend(
self,
level: int,
seed: bytes,
ctx: bytes,
nonce: bytes) -> tuple[list[bytes], list[bool]]:
xof = self.current_xof(
level,
seed,
format_dst(1, 0, 0) + ctx,
nonce,
s = [
bytearray(xof.next(self.KEY_SIZE)),
bytearray(xof.next(self.KEY_SIZE)),
# Use the least significant bits as the control bit correction,
# and then zero it out. This gives effectively 127 bits of
# security, but reduces the number of AES calls needed by 1/3.
t = [bool(s[0][0] & 1), bool(s[1][0] & 1)]
s[0][0] &= 0xFE
s[1][0] &= 0xFE
return ([bytes(s[0]), bytes(s[1])], t)
def convert(
self,
level: int,
seed: bytes,
ctx: bytes,
nonce: bytes) -> tuple[bytes, FieldVec]:
xof = self.current_xof(
level,
seed,
format_dst(1, 0, 1) + ctx,
nonce,
next_seed = xof.next(self.KEY_SIZE)
field = self.current_field(level)
w = xof.next_vec(field, self.VALUE_LEN)
return (next_seed, cast(FieldVec, w))
def current_xof(self,
level: int,
seed: bytes,
dst: bytes,
nonce: bytes) -> Xof:
if level < self.BITS-1:
return XofFixedKeyAes128(seed, dst, nonce)
return XofTurboShake128(seed, dst, nonce)
9.
Security Considerations
VDAFs (
Section 5
) have two essential security goals:
Privacy: an attacker that controls the Collector and a subset of Clients and
a subset of Aggregators learns nothing about the measurements of honest
Clients beyond what it can deduce from the aggregate result. It is assumed
that the attacker controls the entire network except for channels between
honest Clients and honest Aggregators. In particular, it cannot forge or
prevent transmission of messages on these channels.
Verifiability: an attacker that controls a subset of Clients cannot cause
the Collector to compute anything other than the aggregate of the
measurements of honest Clients, plus valid measurements from some of the
attacker-controlled Clients. It is assumed that the attacker eavesdrops on
the network but does not control transmission of messages between honest
parties.
Formal definitions of privacy and verifiability (i.e., robustness) can be found
in
DPRS23
. A VDAF is the core cryptographic primitive of a protocol that
achieves the above privacy and verifiability goals. It is not sufficient on its
own, however. The application will need to assure a few security properties,
for example:
Securely distributing the long-lived parameters, in particular the
verification key.
Establishing secure channels:
Confidential and authentic channels among Aggregators, and between the
Aggregators and the Collector; and
Confidential and Aggregator-authenticated channels between Clients and
Aggregators.
Enforcing the non-collusion properties required of the specific VDAF in use.
In such an environment, a VDAF provides the high-level privacy property
described above: the Collector learns only the aggregate result, and nothing
about individual measurements aside from what can be inferred from the
aggregate result. The Aggregators learn neither individual measurements nor
the aggregate result. The Collector is assured that the aggregate statistic
accurately reflects the inputs as long as the Aggregators correctly executed
their role in the VDAF.
On their own, VDAFs do not provide:
Mitigation of Sybil attacks
Dou02
. In this attack, the adversary
observes a subset of input shares transmitted by a Client it is interested
in. It allows the input shares to be processed, but corrupts and picks bogus
measurements for the remaining Clients. Applications can guard against
these risks by adding additional controls on report submission, such as
Client authentication and rate limits.
Differential privacy
Dwo06
. Depending on the distribution of the
measurements, the aggregate result itself can still leak a significant
amount of information about an individual measurement or the person that
generated it.
Verifiability in the presence of a malicious Aggregator. An Aggregator can,
without detection, manipulate the aggregate result by modifying its own
aggregate share.
Guaranteed output delivery
GSZ20
. An attacker that controls transmission
of messages between honest parties can prevent computation of the aggregate
result by dropping messages.
Security against protocol-confusion attacks. Privacy or verifiability may be
compromised if the attacker coerces an honest party into using either an
incorrect (V)DAF or incorrect parameters for the correct (V)DAF.
9.1.
The Verification Key
The Aggregators are responsible for exchanging the verification key in advance
of executing the VDAF. Any procedure is acceptable as long as the following
conditions are met:
To ensure the computation is verifiably correct, the Aggregators
MUST NOT
reveal the verification key to the Clients. Otherwise, a malicious Client
might be able to exploit knowledge of this key to craft an invalid report
that would be accepted by the Aggregators.
To ensure privacy of the measurements, the Aggregators
MUST
commit to the
verification key prior to processing reports generated by Clients.
Otherwise, the attacker may be able to craft a verification key that, for a
given report, causes an honest Aggregator to leak information about the
measurement during verification.
Meeting these requirements is relatively straightforward. For example, the
Aggregators may designate one of their peers to generate the verification key
and distribute it to the others. To assure Clients of key commitment, the
Clients and (honest) Aggregators
SHOULD
bind the verification key to the
application context. For instance, the "task ID" of DAP
DAP
could be set to
the hash of the verification key; then as long as honest Aggregators only
consume reports for the task indicated by the Client, forging a new key after
the fact would reduce to finding collisions in the underlying hash function.
(Keeping the key secret from the Clients would require the hash function to be
one-way.) However, since rotating the key implies rotating the task ID, this
scheme would not allow key rotation over the lifetime of a task.
9.2.
The Nonce
The sharding and verification phases of VDAF execution depend on a nonce
associated with the Client's report. To ensure privacy of the underlying
measurement, the Client
MUST
generate this nonce using a CSPRNG. This is
required in order to leverage security analysis for the privacy definition of
DPRS23
, which assumes the nonce is chosen at random prior to generating the
report. Uniqueness of the nonce is not sufficient because the verification key
is controlled by the attacker.
Applications will need to protect against replay attacks to prevent disallowed
re-use of reports (see
Section 9.4
). Furthermore, in applications
that aim to achieve differential privacy, it is necessary to limit how much
each party contributes to a single batch or multiple batches. It is
RECOMMENDED
that the nonce generated by the Client be used by the Aggregators for replay
protection.
9.3.
The Public Share
The Aggregators
MUST
ensure they have both received the same public share from
the Client. It is sufficient, for example, to exchange a hash of the public
share over a secure channel.
9.4.
The Aggregation Parameter
As described in
Section 4.3
and
Section 5.3
respectively, DAFs and VDAFs may impose restrictions on the re-use of reports.
For Prio3, reports should only be aggregated once; for Poplar1, reports may be
aggregated multiple times, but never twice at the same level of the tree.
Otherwise, one risks re-using correlated randomness, which might compromise
confidentiality of the Client's measurement.
Higher level applications that use DAFs or VDAFs
MUST
enforce aggregation
parameter validity. In particular, prior to beginning DAF preparation or VDAF
verification with an aggregation parameter provided by the Collector, they
MUST
invoke
is_valid()
to decide if the parameter is valid given the sequence of
previously accepted parameters.
Note that aggregating a batch of reports multiple times, even with a valid
sequence of aggregation parameters, can result in information leakage beyond
what is used by the application.
For example, when Poplar1 is used for heavy hitters, the Aggregators learn not
only the heavy hitters themselves, but also the prefix tree (as defined in
Section 8
) computed along the way. Indeed, this leakage is inherent to any
construction that uses an IDPF (
Section 8.1
) in the same way. Depending on the
distribution of the measurements, the prefix tree can leak a significant amount
of information about unpopular inputs. For instance, it is possible (though
perhaps unlikely) for a large set of non-heavy-hitter values to share a common
prefix, which would be leaked by a prefix tree with a sufficiently small
threshold.
A malicious adversary controlling the Collector and one of the Aggregators can
further turn arbitrary non-heavy prefixes into heavy ones by tampering with the
IDPF output at any position. While the construction ensures that the nodes
evaluated at one level are children of the nodes evaluated at the previous
level, this still may allow an adversary to discover individual non-heavy
strings. This is called a "steering attack".
The only practical, general-purpose defense against steering attacks is to
compose Poplar1 with some mechanism for differential privacy. It is therefore
RECOMMENDED
to use differential privacy for any heavy-hitter type application.
9.5.
Safe Usage of IDPF Outputs
The arithmetic sketch described in
Section 8
is used by the Aggregators to check
that the shares of the vector obtained by evaluating a Client's IDPF at a
sequence of candidate prefixes has at most one non-zero value, and that the
non-zero value is
. Depending on how the values are used, the arithmetic
sketch on its own may not be sufficient to verify the correctness of the
computation. In particular, a malicious Client may attempt to influence the
computation by choosing an IDPF that evaluates to
at more than one node at
a given level of the tree.
This issue can be mitigated by using an IDPF that is extractable as defined
in Appendix D of
BBCGGI21
. Extractability ensures that, for a particular
level of the tree, it is infeasible for an attacker to control values of the
IDPF such that it takes on chosen non-zero values at more than one node. (It
can practically only achieve the zero function, a point function, or a
pseudorandom function.)
The IDPF specified in
Section 8.1
only guarantees extractability at the last level
of the tree. (This is by virtue of using a larger field for the leaves than for
inner nodes and using an XOF to derive leaves that is safe to model as a random
oracle. See
Section 9.6
.) For intermediate levels, it is feasible for a
client to produce IDPF shares with two controlled non-zero nodes.
This is not an issue for running heavy hitters, since (1) each node in the
prefix tree is a child of a previously traversed node, (2) the arithmetic sketch
would detect double voting at every level of the prefix tree, and (3) the IDPF
is extractable at the last level of the tree. However, the lack of
extractability at intermediate levels may result in attacks on the correctness of
the computation in certain applications.
Thus applications
SHOULD NOT
use prefix counts for intermediate levels for any
purpose beyond computing the prefix tree for heavy hitters.
9.6.
Safe Usage of XOFs
In the security analyses of these protocols, XOFs (
Section 6.2
) are usually modeled
as random oracles. XofTurboShake128 is designed to be indifferentiable from a
random oracle
MRH04
, making it a suitable choice for most situations.
The one exception is the IDPF of
Section 8.3
. Here, a random oracle is not
needed to prove privacy, since the analysis of
BBCGGI21
, Proposition 1, only
requires a Pseudorandom Generator (PRG). As observed in
GKWY20
, a PRG can be
instantiated from a correlation-robust hash function
. Informally,
correlation robustness requires that for a random
H(xor(r, x))
is
computationally indistinguishable from a random function of
. A PRG can
therefore be constructed as
PRG(r) = H(xor(r, 1)) || H(xor(r, 2)) || ...
since each individual hash function evaluation is indistinguishable from a random
function.
XofFixedKeyAes128 in
Section 6.2.2
implements a correlation-robust
hash function using fixed-key AES. For security, it assumes that AES with a
fixed key can be modeled as a random permutation
GKWY20
. Additionally, a
different AES key is used for every report, which in the ideal cipher model
leads to better concrete security
GKWWY20
Note that for verifiability, the analysis of
BBCGGI21
still assumes a
random oracle to make the IDPF extractable. Thus XofTurboShake128 is used
instead for the last level of the tree. It is important that XofTurboShake128
supports many seed lengths, in particular 16 bytes, as this is the seed size
for the inner levels.
While XofFixedKeyAes128 has been shown to be differentiable from a random
oracle
GKWWY20
, there are no known attacks exploiting this difference. And
even if the IDPF is not extractable, Poplar1 guarantees that every client can
contribute to at most one prefix among the ones being evaluated by the helpers.
9.7.
Choosing FLP Parameters
Prio3 and other systems built from the FLP of
Section 7.3
may benefit from
choosing a field size that is as small as possible. Generally speaking, a
smaller field results in lower communication and storage costs. Care must be
taken, however, since a smaller field also results in degraded (or even
vacuous) verifiability.
Different variants of Prio3 (
Section 7
) use different field sizes: Prio3Count
and Prio3Sum use Field64; but Prio3SumVec, Prio3Histogram, and
Prio3MultihotCountVec all use Field128, a field that is twice as large as
Field64. This is due to the use of joint randomness (
Section 7.1
) in the latter
variants. Joint randomness allows for more flexible circuit design (see
Section 7.3.1.1
), but opens up Prio3 to offline attacks in
which the attacker searches for input shares for an invalid measurement that
derive joint randomness that causes the circuit to accept. Choosing a large
enough field ensures this computation is too expensive to be feasible. (See
DPRS23
, Theorem 1.) Note that privacy is not susceptible to such attacks.
Another way to mitigate this issue (or improve verifiability in general) is to
generate and verify multiple, independent proofs. (See
Section 7.1.2
.) For
Prio3, the
PROOFS
parameter controls the number of proofs (at least one) that
are generated and verified. In general the soundness error of the FLP is given
by the following formula:
(circuit_soundness + flp_soundness)**PROOFS
where:
circuit_soundness
is the soundness of the validity circuit
Section 7.3.2
flp_soundness
is the base soundness of the proof system (
BBCGGI19
Theorem 4.3)
For circuits involving joint randomness, one should aim for the soundness error
to be close to
2**-128
in order to mitigate offline attacks. Such circuits
MUST
use Field128 with at least one proof or Field64 with at least three
proofs. Depending on the circuit, Field64 with two proofs might have
significantly lower soundness than Field128 with one proof.
Weak parameters (too small a field, too few proofs, or both) can be exploited
to attack any aggregation task using those parameters. To mitigate offline
attacks, it is necessary to disable all tasks that use the weak parameters.
9.8.
Choosing the Number of Aggregators
Two Aggregators are required for privacy in the threat model, but some (V)DAFs,
including Prio3 (
Section 7
), allow for any number of Aggregators, only one of
which needs to be trusted in order for the computation to be private. To hedge
against corruptions that happen during the course of the attack, deployments
may consider involving more than two Aggregators as described for example in
Section 5.7.2
. Note however that some schemes are not compatible with this mode
of operation, such as Poplar1.
9.9.
Defense-in-Depth Measures
Prio3 and Poplar1 are designed to resist some attacks that fall outside the
main threat model for VDAFs.
Broadly speaking, domain separation is used to prevent cross protocol attacks,
in which data from evaluation of one VDAF translates to an attack against another.
For example:
Weak entropy sources: the VDAF algorithm ID is bound to each XOF invocation,
thereby ensuring the outputs are different between VDAF invocations, even if
the underlying randomness is the same. For example, two different instances
of Prio3 would compute different measurement shares.
Weak parameters: Prio3 variants that require joint randomness are subject to
offline attacks against verifiability. These attacks are feasible if the field
size or number of proofs is sufficiently small. (See
Section 9.7
.) The joint randomness derivation is bound to both
the field (via the algorithm ID) and the number of proofs, thereby ensuring
that joint randomness derived for weak parameters is not reused for stronger
parameters. In addition, the joint randomness is bound to the application
context, meaning any work the attacker does to attack some application is
not useful for other applications that use the same parameters.
There are also some important limitations to be aware of. For example, Prio3
provides domain separation between families of circuits, but does not provide
domain separation between instances of a circuit. Concretely, it is possible
for Aggregators to accept a report for Prio3SumVec from a Client who disagrees
with them on the value of
max_measurement
and
length
(so long as the encoded
measurement is the same size). This is because there is no
binding of the circuit parameters to the computation.
9.10.
Side-Channel Resistance
Implementations of VDAFs should incorporate defenses against side-channel
attacks. For side-channel attacks against the privacy security goal, the
relevant threat model includes an attacker that may control the Collector, a
subset of Clients, and a subset of Aggregators, and monitor side-channel
signals from the honest Clients and Aggregators. Side-channel attacks by third
parties may indirectly target verifiability by trying to leak the Aggregators'
verification key. Thus, implementations of Clients and Aggregators should treat
measurements, input shares, output shares, and the verification key as secret,
and avoid leaking those secret values or any intermediate computations that
depend on them.
For example, the following routines should all be implemented in a side-channel
resistant manner.
Finite field arithmetic
XOFs
IDPF generation and evaluation, including handling of control bits
10.
IANA Considerations
IANA is requested to make one new registry:
DAF and VDAF Identifiers
This registry should be created under the heading "Verifiable Distributed
Aggregation Functions (VDAF)", and administered under the Specification Required
policy
RFC8126
The "VDAF Identifiers" registry lists identifiers for Distributed Aggregation
Functions (DAFs) and Verifiable Distributed Aggregation Functions (VDAFs). These
identifiers are four-byte values, so the minimum possible value is
0x00000000
and the maximum possible value is
0xffffffff
Template:
Value: The four-byte identifier for the DAF or VDAF
Scheme: The name of the DAF or VDAF
Type: Either "DAF" for a Distributed Aggregation Function or "VDAF" for a
Verifiable Distributed Aggregation Function
Reference: Where the algorithm is defined
The initial contents of the registry are as follows:
Table 19
Verifiable Distributed Aggregation Function Identifiers Registry
Value
Scheme
Type
Reference
0x00000000
Reserved
n/a
RFC XXXX
0x00000001
Prio3Count
VDAF
Section 7.4.1
of RFC XXXX
0x00000002
Prio3Sum
VDAF
Section 7.4.2
of RFC XXXX
0x00000003
Prio3SumVec
VDAF
Section 7.4.3
of RFC XXXX
0x00000004
Prio3Histogram
VDAF
Section 7.4.4
of RFC XXXX
0x00000005
Prio3MultihotCountVec
VDAF
Section 7.4.5
of RFC XXXX
0x00000006
Poplar1
VDAF
Section 8.2
of RFC XXXX
0xFFFF0000
to
0xFFFFFFFF
Reserved for Private Use
n/a
n/a
(RFC EDITOR: Please replace "RFC XXXX" above with the RFC number assigned to
this document.)
VDAF identifiers are used for domain separation, as described in
Section 6.2.3
. Domain separation guards against cross protocol attacks and
certain failures of entropy sources. See
Section 9.9
The benefits of domain separation are undermined if different VDAFs are used
with the same VDAF Identifier. The "Reserved for Private Use" code points
should thus be used judiciously, because they provide no defense against such
collisions. Applications
SHOULD
prefer the use of registered code points.
11.
References
11.1.
Normative References
[AES]
Dworkin, M. J.
Barker, E.
Nechvatal, J. R.
Foti, J.
Bassham, L. E.
Roback, E.
, and
J. Dray Jr
"Advanced Encryption Standard (AES)"
2001
[RFC2119]
Bradner, S.
"Key words for use in RFCs to Indicate Requirement Levels"
BCP 14
RFC 2119
DOI 10.17487/RFC2119
March 1997
[RFC8126]
Cotton, M.
Leiba, B.
, and
T. Narten
"Guidelines for Writing an IANA Considerations Section in RFCs"
BCP 26
RFC 8126
DOI 10.17487/RFC8126
June 2017
[RFC8174]
Leiba, B.
"Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words"
BCP 14
RFC 8174
DOI 10.17487/RFC8174
May 2017
[RFC8446]
Rescorla, E.
"The Transport Layer Security (TLS) Protocol Version 1.3"
RFC 8446
DOI 10.17487/RFC8446
August 2018
[RFC9861]
Viguier, B.
Wong, D., Ed.
Van Assche, G., Ed.
Dang, Q., Ed.
, and
J. Daemen, Ed.
"KangarooTwelve and TurboSHAKE"
RFC 9861
DOI 10.17487/RFC9861
October 2025
11.2.
Informative References
[AGJOP21]
Addanki, S.
Garbe, K.
Jaffe, E.
Ostrovsky, R.
, and
A. Polychroniadou
"Prio+: Privacy Preserving Aggregate Statistics via Boolean Shares"
Security and Cryptography for Networks (SCN)
2022
[BBCGGI19]
Boneh, D.
Boyle, E.
Corrigan-Gibbs, H.
Gilboa, N.
, and
Y. Ishai
"Zero-Knowledge Proofs on Secret-Shared Data via Fully Linear PCPs"
Crypto
2019
[BBCGGI21]
Boneh, D.
Boyle, E.
Corrigan-Gibbs, H.
Gilboa, N.
, and
Y. Ishai
"Lightweight Techniques for Private Heavy Hitters"
IEEE Security & Privacy (S&P)
2021
[BBCGGI23]
Boneh, D.
Boyle, E.
Corrigan-Gibbs, H.
Gilboa, N.
, and
Y. Ishai
"Arithmetic Sketching"
Crypto
2023
[BGI15]
Boyle, E.
Gilboa, N.
, and
Y. Ishai
"Function Secret Sharing"
Eurocrypt
2015
[CGB17]
Boneh, D.
and
H. Corrigan-Gibbs
"Prio: Private, Robust, and Scalable Computation of Aggregate Statistics"
USENIX Symposium on Networked Systems Design and Implementation (NSDI)
2017
[DAP]
Geoghegan, T.
Patton, C.
Pitman, B.
Rescorla, E.
, and
C. A. Wood
"Distributed Aggregation Protocol for Privacy Preserving Measurement"
Work in Progress
Internet-Draft, draft-ietf-ppm-dap-16
2 September 2025
[Dou02]
Douceur, J. R.
"The Sybil Attack"
International Workshop on Peer-to-Peer Systems (IPTPS)
2002
[DPRS23]
Hannah Davis
Christopher Patton
Mike Rosulek
, and
Phillipp Schoppmann
"Verifiable Distributed Aggregation Functions"
Privacy Enhancing Technologies Symposium (PETS)
2023
[Dwo06]
Cynthia Dwork
"Differential Privacy"
International Colloquium on Automata, Languages, and Programming (ICALP)
2006
[ENPA]
"Exposure Notification Privacy-preserving Analytics (ENPA) White Paper"
2021
[EPK14]
Erlingsson, Ú.
Pihur, V.
, and
A. Korolova
"RAPPOR: Randomized Aggregatable Privacy-Preserving Ordinal Response"
ACM Conference on Computer and Communications Security (CCS)
2014
[Faz25]
Faz-Hernandez, A.
"Rhizomes and the Roots of Efficiency—Improving Prio"
Springer Nature Switzerland
Lecture Notes in Computer Science pp. 425-449
DOI 10.1007/978-3-032-06754-8_16
ISBN ["9783032067531", "9783032067548"]
October 2025
[GI14]
Gilboa, N.
and
Y. Ishai
"Distributed Point Functions and Their Applications"
Eurocrypt
2014
[GKWWY20]
Guo, C.
Katz, J.
Wang, X.
Weng, C.
, and
Y. Yu
"Better concrete security for half-gates garbling (in the multi-instance setting)"
Crypto
2020
[GKWY20]
Guo, C.
Katz, J.
Wang, X.
, and
Y. Yu
"Efficient and Secure Multiparty Computation from Fixed-Key Block Ciphers"
IEEE Security & Privacy (S&P)
2020
[GSZ20]
Goyal, V.
Song, Y.
, and
C. Zhu
"Guaranteed Output Delivery Comes Free in Honest Majority MPC"
Crypto
2020
[I-D.draft-irtf-cfrg-cryptography-specification-02]
Sullivan, N.
and
C. A. Wood
"Guidelines for Writing Cryptography Specifications"
Work in Progress
Internet-Draft, draft-irtf-cfrg-cryptography-specification-02
7 July 2025
[MPDST25]
Mouris, D.
Patton, C.
Davis, H.
Sarkar, P.
, and
N. G. Tsoutsos
"Mastic: Private Weighted Heavy-Hitters and Attribute-Based Metrics"
Privacy Enhancing Technologies Symposium (PETS)
2025
[MPRV09]
Mironov, I.
Pandey, O.
Reingold, O.
, and
S. Vadhan
"Computational Differential Privacy"
Crypto
2009
[MRH04]
Maurer, U.
Renner, R.
, and
C. Holenstein
"Indifferentiability, impossibility results on reductions, and applications to the random oracle methodology"
Theory of Cryptography (TCC)
2004
[OriginTelemetry]
"Origin Telemetry"
2020
[PANEL-FEEDBACK]
Hesse, J.
"Review of draft-irtf-cfrg-vdaf-15"
2025
[Pol71]
Pollard, J.
"The fast Fourier transform in a finite field"
American Mathematical Society (AMS)
Mathematics of Computation vol. 25, no. 114, pp. 365-374
DOI 10.1090/s0025-5718-1971-0301966-0
1971
[SML24]
Satriawan, A.
Mareta, R.
, and
H. Lee
"A Complete Beginner Guide to the Number Theoretic Transform (NTT)"
IEEE Access, vol. 11
2024
[TestVectors]
"Test vectors for Prio3 and Poplar1"
commit hash 5b7df1d
December 2024
Appendix A.
FLP Gadgets
This section defines gadgets used to specify variants of Prio3 defined in
Section 7.4
and to construct FLPs as defined in
Section 7.3
A.1.
Multiplication
The multiplication gadget takes in a pair of inputs and multiplies them
together. The degree of this circuit is
class Mul(Gadget[F]):
ARITY = 2
DEGREE = 2
def eval(self, _field: type[F], inp: list[F]) -> F:
return inp[0] * inp[1]
def eval_poly(self,
field: type[F],
inp_poly: list[list[F]]) -> list[F]:
lag = Lagrange(field)
return lag.poly_mul(inp_poly[0], inp_poly[1])
A.2.
Polynomial Evaluation
The polynomial-evaluation gadget takes in a single input
and returns
p(x)
, where
is a polynomial specified by the user. Its degree is the same
as
class PolyEval(Gadget[F]):
ARITY = 1
p: list[int] # polynomial coefficients
def __init__(self, p: list[int], num_calls: int):
"""
Instantiate this gadget with the given polynomial.
"""
# Strip leading zeros.
for i in reversed(range(len(p))):
if p[i] != 0:
p = p[:i+1]
break
if len(p) < 1:
raise ValueError('invalid polynomial: zero length')
self.p = p
self.DEGREE = len(p) - 1
wire_poly_length = wire_poly_len(num_calls)
gadget_poly_length = gadget_poly_len(
self.DEGREE,
wire_poly_length,
self.n = next_power_of_2(gadget_poly_length)
def eval(self, field: type[F], inp: list[F]) -> F:
p = [field(coeff) for coeff in self.p]
return poly_eval(field, p, inp[0])
def eval_poly(self,
field: type[F],
inp_poly: list[list[F]]) -> list[F]:
inp_poly_len = len(inp_poly[0])
assert_power_of_2(inp_poly_len)
# Convert the input polynomial from Lagrange to monomial basis.
inp_mon = field.inv_ntt(inp_poly[0], inp_poly_len)
# Obtain n evaluations of the input polynomial I.
inp_lag = field.ntt(inp_mon, self.n)
# Returns the polynomial composition (P*I)
p_mon = [field(coeff) for coeff in self.p]
return [poly_eval(field, p_mon, x) for x in inp_lag]
A.3.
Parallel Sum
The parallel-sum gadget applies a subcircuit to multiple inputs in parallel,
then returns the sum of the results. The arity of the gadget is equal to
count
times the arity of the subcircuit, where
count
denotes the number of times the
subcircuit is called. The degree of the gadget is equal to the degree of the
subcircuit.
class ParallelSum(Gadget[F]):
subcircuit: Gadget[F]
count: int
def __init__(self, subcircuit: Gadget[F], count: int):
self.subcircuit = subcircuit
self.count = count
self.ARITY = subcircuit.ARITY * count
self.DEGREE = subcircuit.DEGREE
def eval(self, field: type[F], inp: list[F]) -> F:
out = field(0)
for i in range(self.count):
start_index = i * self.subcircuit.ARITY
end_index = (i + 1) * self.subcircuit.ARITY
out += self.subcircuit.eval(
field,
inp[start_index:end_index],
return out
def eval_poly(self,
field: type[F],
inp_poly: list[list[F]]) -> list[F]:
output_poly_length = next_power_of_2(gadget_poly_len(
self.DEGREE,
len(inp_poly[0]),
))
out_sum = [field(0) for _ in range(output_poly_length)]
for i in range(self.count):
start_index = i * self.subcircuit.ARITY
end_index = (i + 1) * self.subcircuit.ARITY
out_current = self.subcircuit.eval_poly(
field,
inp_poly[start_index:end_index]
for j in range(output_poly_length):
out_sum[j] += out_current[j]
return out_sum
A.4.
Shims for Generating and Querying Proofs
This section specifies two "shim" gadgets, one for generating an FLP as
specified in
Section 7.3.3
and another for querying an FLP
as specified in
Section 7.3.4
class ProveGadget(Gadget[F]):
"""
Gadget wrapper that records the input wires for each evaluation.
"""
def __init__(self,
field: type[F],
wire_seeds: list[F],
g: Gadget[F],
g_calls: int):
p = wire_poly_len(g_calls)
self.inner = g
self.ARITY = g.ARITY
self.DEGREE = g.DEGREE
self.wires = []
self.k = 0 # evaluation counter
for s in wire_seeds:
wire = field.zeros(p)
wire[0] = s # set the wire seed
self.wires.append(wire)
def eval(self, field: type[F], inp: list[F]) -> F:
self.k += 1
for j in range(len(inp)):
self.wires[j][self.k] = inp[j]
return self.inner.eval(field, inp)
def eval_poly(self,
field: type[F],
inp_poly: list[list[F]]) -> list[F]:
return self.inner.eval_poly(field, inp_poly)
@classmethod
def wrap(cls,
valid: Valid[Measurement, AggResult, F],
prove_rand: list[F],
) -> Valid[Measurement, AggResult, F]:
"""
Make a copy of `valid` with each gadget wrapped for recording
the wire inputs. `prove_rand` is used to produce the wire
seeds for each gadget.
"""
wrapped_gadgets: list[Gadget[F]] = []
for (g, g_calls) in zip(valid.GADGETS, valid.GADGET_CALLS):
(wire_seeds, prove_rand) = front(g.ARITY, prove_rand)
wrapped = cls(valid.field, wire_seeds, g, g_calls)
wrapped_gadgets.append(wrapped)
wrapped_valid = deepcopy(valid)
wrapped_valid.GADGETS = wrapped_gadgets
return wrapped_valid
class QueryGadget(Gadget[F]):
"""
Gadget wrapper that records the input wires for each evaluation.
Rather than evaluate the circuit, use the provided gadget
polynomial to produce the output.
"""
def __init__(
self,
field: type[F],
wire_seeds: list[F],
gadget_poly: list[F],
g: Gadget[F],
g_calls: int):
p = wire_poly_len(g_calls)
self.alpha = field.gen() ** (field.GEN_ORDER // p)
self.ARITY = g.ARITY
self.DEGREE = g.DEGREE
self.wires = []
self.k = 0
for s in wire_seeds:
wire = field.zeros(p)
wire[0] = s # set the wire seed
self.wires.append(wire)
# Recover all the values of the gadget_poly.
lag = Lagrange(field)
n = next_power_of_2(len(gadget_poly))
gadget_poly = list(gadget_poly)
lag.extend_values_to_power_of_2(gadget_poly, n)
# Calculate 'size' evaluations of the gadget_poly.
size = next_power_of_2(gadget_poly_len(g.DEGREE, p))
while len(gadget_poly) < size:
gadget_poly = lag.double_evaluations(gadget_poly)
self.poly = gadget_poly
# Get the step size used to index the gadget evaluations.
log_size = assert_power_of_2(size)
log_p = assert_power_of_2(p)
self.step = 1 << (log_size-log_p)
def eval(self, field: type[F], inp: list[F]) -> F:
self.k += 1
for j in range(len(inp)):
self.wires[j][self.k] = inp[j]
return self.poly[self.k*self.step]
@classmethod
def wrap(cls,
valid: Valid[Measurement, AggResult, F],
proof: list[F]) -> Valid[Measurement, AggResult, F]:
wrapped_gadgets: list[Gadget[F]] = []
for (g, g_calls) in zip(valid.GADGETS, valid.GADGET_CALLS):
p = wire_poly_len(g_calls)
g_poly_len = gadget_poly_len(g.DEGREE, p)
(wire_seeds, proof) = front(g.ARITY, proof)
(gadget_poly, proof) = front(g_poly_len, proof)
wrapped = cls(valid.field,
wire_seeds,
gadget_poly,
g,
g_calls)
wrapped_gadgets.append(wrapped)
wrapped_valid = deepcopy(valid)
wrapped_valid.GADGETS = wrapped_gadgets
return wrapped_valid
Appendix B.
VDAF Verification State
This section lists the classes used to define each Aggregator's state during
VDAF verification (
Section 5.7
).
class State:
pass
class Start(State):
pass
class Continued(State, Generic[VerifyState]):
def __init__(self,
verify_state: VerifyState,
verify_round: int,
outbound: bytes):
self.verify_state = verify_state
self.verify_round = verify_round
self.outbound = outbound
def __eq__(self, other: object) -> bool:
return isinstance(other, Continued) and \
self.verify_state == other.verify_state and \
self.verify_round == other.verify_round and \
self.outbound == other.outbound
class Finished(State, Generic[OutShare]):
def __init__(self, out_share: OutShare):
self.out_share = out_share
def __eq__(self, other: object) -> bool:
return isinstance(other, Finished) and \
self.out_share == other.out_share
class FinishedWithOutbound(State, Generic[OutShare]):
def __init__(self, out_share: OutShare, outbound: bytes):
self.out_share = out_share
self.outbound = outbound
def __eq__(self, other: object) -> bool:
return isinstance(other, FinishedWithOutbound) and \
self.out_share == other.out_share and \
self.outbound == other.outbound
class Rejected(State):
pass
Appendix C.
Test Vectors
Test vectors for Prio3 (
Section 7
) and Poplar1 (
Section 8
) are available at
TestVectors
. The test vector directory,
test_vec/vdaf
, contains a set of
JSON files. Each file contains a test vector for an instance of class
Vdaf
as
defined in
Section 5
. A test vector covers sharding, verification, aggregation,
and unsharding of a batch of several measurements. The test vector schema is
defined below.
C.1.
Schema
ctx
The application context string encoded in hexadecimal.
verify_key
The verification key encoded in hexadecimal.
agg_param
The aggregation parameter encoded in hexadecimal.
reports
A list of objects with the following schema:
measurement
The measurement of type
Measurement
defined by the VDAF.
nonce
The nonce encoded in hexadecimal.
rand
The sharding randomness encoded in hexadecimal.
public_share
The expected public share encoded in hexadecimal.
input_shares
The expected list of input shares, each encoded in hexadecimal.
verifier_shares
The expected list of verifier shares generated by each Aggregator at each
round of verification, encoded in hexadecimal.
verifier_messages
The expected list of verifier messages for each round of
verification, encoded in hexadecimal.
out_shares
The expected list of output shares, encoded in hexadecimal.
agg_shares
The expected aggregate shares encoded in hexadecimal.
agg_result
The expected aggregate result of type
AggResult
defined by the VDAF.
operations
This lists the VDAF operations that should be executed as part of known
answer tests, using messages from this test vector as input. Operations
should be executed in the order they appear, to ensure that verification state
values are computed before they are consumed. Verification state values are
not included in test vectors because this document does not specify their
representation or encoding.
Each operation in the
operations
list has the following schema:
operation
The type of operation to be performed. This is one of "shard", "verify_init",
"verifier_shares_to_message", "verify_next", "aggregate", or "unshard".
round
For any verification operation, the round number of the operation to be
performed. This determines which verifier share, verification state, and/or
verifier message to use.
aggregator_id
The aggregator ID to use when performing this operation. This determines
which messages and which verification state to use, in addition to the
aggregator ID argument itself.
report_index
The index of the report on which to perform this operation. This is an index
into the
reports
array.
success
If this is
True
, the operation should succeed, and its output should match
the corresponding values in the test vector. If this is
False
, the operation
should fail, terminating verification of this report.
The test vector schema also includes whatever parameters are required to
instantiate the VDAF. These are listed in the subsections below.
C.1.1.
Prio3Count
shares
The number of shares, an integer.
C.1.2.
Prio3Sum
shares
The number of shares, an integer.
max_measurement
The largest valid measurement, an integer. Each measurement is in the range
[0, max_measurement]
C.1.3.
Prio3SumVec
shares
The number of shares, an integer.
length
The length of the vector, an integer.
chunk_length
a parameter of the ParallelSum gadget, an integer.
max_measurement
The largest allowable value of each element of the vector, an integer. Each
element is in the range
[0, max_measurement]
C.1.4.
Prio3Histogram
shares
The number of shares, an integer.
length
The length of the vector, an integer.
chunk_length
a parameter of the ParallelSum gadget, an integer.
C.1.5.
Prio3MultihotCountVec
shares
The number of shares, an integer.
length
The length of the vector, an integer.
chunk_length
a parameter of the ParallelSum gadget, an integer.
max_weight
The largest vector weight, an integer. The sum of the elements of the
measurement must be in the range
[0, max_weight]
C.1.6.
Poplar1
bits
The length of each input in bits, an integer.
Acknowledgments
The impetus of this work is the success of recent deployments of predecessors
of Prio3. These include the Mozilla Origin Telemetry project
OriginTelemetry
and the Exposure Notification Private Analytics system
ENPA
developed jointly by ISRG, Google, Apple, and others. Together these
systems have aggregated data from hundreds of millions of users.
As the name implies, Prio3 is a descendant of the original Prio construction
CGB17
. A second iteration was deployed in the
ENPA
system, and like the
VDAF described here, the ENPA system was built from techniques introduced in
BBCGGI19
that significantly improve communication cost. That system was
specialized for a particular aggregation function; the goal of Prio3 is to
provide the same level of generality as the original construction.
The security considerations in
Section 9
are based largely on the security
analysis of
DPRS23
. Thanks to Hannah Davis and Mike Rosulek, who lent their
time to developing definitions and security proofs.
Thanks to Julia Hesse who provided feedback on behalf of the Crypto Review
Panel.
Special thanks to Armando Faz-Hernandez who discovered a number of algorithmic
improvements for FLPs
Faz25
that significantly improved the runtime of
Prio3 and other applications. Also, thanks to Ian Goldberg for pointing out the
bit decomposition technique used in the range proofs of the Prio3 variants.
Thanks to Junye Chen, Henry Corrigan-Gibbs, Simon Friedberger, Tim Geoghegan,
Albert Liu, Brandon Pitman, Mariana Raykova, Michael Rosenberg, Jacob
Rothstein, Shan Wang, Xiao Wang, Bas Westerbaan, and Christopher Wood for
useful feedback on and contributions to the spec.
Authors' Addresses
Richard L. Barnes
Cisco
Email:
rlb@ipv.sx
David Cook
ISRG
Email:
divergentdave@gmail.com
Christopher Patton
Cloudflare
Email:
chrispatton+ietf@gmail.com
Phillipp Schoppmann
Google
Email:
schoppmann@google.com
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Richard Barnes
David Cook
Christopher Patton
Phillipp Schoppmann
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