Lattice-Based Post-quantum iO from Circular Security with Random Opening Assumption | Springer Nature Link

Lattice-Based Post-quantum iO from Circular Security with Random Opening Assumption | Springer Nature Link
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Lattice-Based Post-quantum iO from Circular Security with Random Opening Assumption
Conference paper
First Online:
17 August 2025
pp 3–38
Cite this conference paper
Advances in Cryptology – CRYPTO 2025
(CRYPTO 2025)
Abstract
Indistinguishability obfuscation (
\(\textsf{iO}\)
) stands out as a powerful cryptographic primitive but remains notoriously difficult to realize under simple-to-state, post-quantum assumptions. Recent works have proposed lattice-inspired
\(\textsf{iO}\)
constructions backed by new “LWE-with-hints” assumptions, which posit that certain distributions of LWE samples retain security despite auxiliary information. However, subsequent cryptanalysis has revealed structural vulnerabilities in these assumptions, leaving us without any post-quantum
\(\textsf{iO}\)
candidates supported by simple, unbroken assumptions.
Motivated by these proposals, we introduce the
Circular Security with Random Opening
\({\textsf{CRO}}\)
) assumption—a new LWE-with-hint assumption that addresses structural weaknesses from prior assumptions, and based on our systematic examination, does not appear vulnerable to known cryptanalytic techniques. In
\({\textsf{CRO}}\)
, the hints are random “openings” of zero-encryptions under the Gentry–Sahai–Waters (GSW) homomorphic encryption scheme. Crucially, these zero-encryptions are efficiently derived from the original LWE samples via a special, carefully designed procedure, ensuring that the openings are marginally random. Moreover, the openings do not induce any natural leakage on the LWE noises. These two features—
marginally random hints and the absence of (natural) noise leakage
—rule out important classes of attacks that had undermined all previous LWE-with-hint assumptions for
\(\textsf{iO}\)
. Therefore, our new lattice-based assumption for
\(\textsf{iO}\)
provides a qualitatively different target for cryptanalysis compared to existing assumptions.
To build
\(\textsf{iO}\)
under this less-structured
\({\textsf{CRO}}\)
assumption, we develop several new technical ideas. In particular, we devise an
oblivious LWE sampling
procedure, which succinctly encodes random LWE secrets and smudging noises, and uses a tailored-made homomorphic evaluation procedure to generate secure LWE samples. Crucially, all non-LWE components in this sampler, including the secrets and noises of the generated samples, are independently and randomly distributed, avoiding attacks on non-LWE components.
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Notes
1.
Otherwise, a trivial construction would be to simply output the truth table as the obfuscated circuit.
2.
Sometimes more than one LWE secrets are involved and the key-dependent messages may depend on multiple secrets.
3.
Private-coin evasive LWE assumptions are pseudorandomness type assumptions. However, they only enable pseudorandom functionalities.
4.
One can always append zeros to the function output to make
a multiple of
\((n+1){\lceil \log q\rceil }\)
. This is a technicality due to the interface of GSW that we formalized in accordance with the abstract definition of HE, requiring that when encrypting a
\({\mathbb {Z}}_q\)
vector, the length of the vector is
\((n+1){\lceil \log q\rceil }\)
5.
The formal description of the
\(f^\textrm{circ}\)
-circular assumption, along with the proofs of the two theorems, are provided in the full version [
49
].
6.
Functional encoding, introduced by [
71
], is an intermediate primitive implying
\(\textsf{xiO}\)
. Due to space constraints, we refer the readers to the full version [
49
] for the definition.
7.
This should be separated from LWE-based constructions, e.g., NIZK, where pseudorandomness does not hold but ZK or indistinguishability holds. Such behaviors are the result of careful design, whereas when formulating assumptions, we are considering LWE encodings that we do not fully know how to analyze.
8.
We thank the anonymous reviewer for pointing out this observation.
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Acknowledgment
The authors would like to thank Hoeteck Wee for many insightful discussions about evasive LWE assumptions, and attacks on assumptions underlying lattice-based
\(\textsf{iO}\)
and
\(\textsf{PrO}\)
candidates. The authors also would like to thank the anonymous reviewers for insightful observations and comments. Yao Ching Hsieh and Huijia Lin were supported by NSF grant CNS-2026774, and a Simons Collaboration on the Theory of Algorithmic Fairness. Aayush Jain was supported by a Google Faculty Research Scholar 2023, a Stellar Foundation Grant, CYLAB of CMU, and an NSF CAREER CNS-2441647.
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Carnegie Mellon University, Pittsburgh, USA
Aayush Jain
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Hsieh, YC., Jain, A., Lin, H. (2025). Lattice-Based Post-quantum iO from Circular Security with Random Opening Assumption.

In: Tauman Kalai, Y., Kamara, S.F. (eds) Advances in Cryptology – CRYPTO 2025. CRYPTO 2025. Lecture Notes in Computer Science, vol 16006. Springer, Cham. https://doi.org/10.1007/978-3-032-01907-3_1
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