International Association
for Cryptologic Research

Noisy linear algebraic assumptions with respect to random matrices, in particular Learning with Errors ($\LWE$) and Alekhnovich Learning Parity with Noise (Alekhnovich $\LPN$), are among the most investigated assumptions that imply post-quantum public-key encryption (PKE). They enjoy elegant mathematical structure. Indeed, efforts to build post-quantum PKE and advanced primitives such as homomorphic encryption and indistinguishability obfuscation have increasingly focused their attention on these two assumptions and their variants. Unfortunately, this increasing reliance on these two assumptions for building post-quantum cryptography leaves us vulnerable to potential quantum (and classical) attacks on Alekhnovich $\LPN$ and $\LWE$. Quantum algorithms is a rapidly advancing area, and we must stay prepared for unexpected cryptanalytic breakthroughs. Just three decades ago, a short time frame in the development of our field, Shor's algorithm rendered most then-popular number theoretic and algebraic assumptions quantumly broken. Furthermore, within the last several years, we have witnessed major classical and quantum breaks on several assumptions previously introduced for post-quantum cryptography. Therefore, we ask the following question: \begin{center} \emph{In a world where both $\LWE$ and Alekhnovich $\LPN$ are broken, can there still exist noisy linear assumptions that remain plausibly quantum hard and imply PKE?} \end{center} To answer this question positively, we introduce two natural noisy-linear algebraic assumptions that are both with respect to random matrices, exactly like $\LWE$ and Alekhnovich $\LPN$, but with different error distributions. Our error distribution combines aspects of both small norm and sparse error distributions. We design a PKE from these assumptions and give evidence that these assumptions are likely to still be secure even in a world where both the $\LWE$ and Alekhnovich $\LPN$ assumptions are simultaneously broken. We also study basic properties of these assumptions, and show that in the parameter settings we employ to build PKE, neither of them are ``lattice'' assumptions in the sense that we don't see a way to attack them using a lattice closest vector problem solver, except via $\NP$-completeness reductions.

To date, the strongest notions of security achievable for two-round publicly-verifiable cryptographic proofs for NP are witness indistinguishability (Dwork-Naor 2000, Groth-Ostrovsky-Sahai 2006), witness hiding (Bitansky-Khurana-Paneth 2019, Kuykendall-Zhandry 2020), and super-polynomial simulation (Pass 2003, Khurana-Sahai 2017). On the other hand, zero-knowledge and even weak zero-knowledge (Dwork-Naor-Reingold-Stockmeyer 1999) are impossible in the two-round publicly-verifiable setting (Goldreich-Oren 1994). This leaves an enormous gap in our theoretical understanding of known achievable security and the impossibility results for two-round publicly-verifiable cryptographic proofs for NP. Towards filling this gap, we propose a new and natural notion of security, called witness semantic security, that captures the natural and strong notion that an adversary should not be able to learn any partial information about the prover's witness beyond what it could learn given only the statement x. Not only does our notion of witness semantic security subsume both witness indistinguishability and witness hiding, but it also has an easily appreciable interpretation. Moreover, we show that assuming the subexponential hardness of LWE, there exists a two-round public-coin publicly-verifiable witness semantic secure argument. To our knowledge, this is the strongest form of security known for this setting. As a key application of our work, we show that non-interactive zero-knowledge (NIZK) arguments in the common reference string (CRS) model can additionally maintain witness semantic security even when the CRS is maliciously generated. Our work gives the first construction from (subexponential) standard assumptions that achieves a notion stronger than witness-indistinguishability against a malicious CRS authority. In order to achieve our results, we give the first construction of a ZAP from subexponential LWE that is adaptively sound. Additionally, we propose a notion of simulation using non-uniform advice about a malicious CRS, which we also believe will be of independent interest.

Indistinguishability Obfuscation (iO) is a highly versatile primitive implying a myriad advanced cryptographic applications. Up until recently, the state of feasibility of iO was unclear, which changed with works (Jain-Lin-Sahai STOC 2021, Jain-Lin-Sahai Eurocrypt 2022) showing that iO can be finally based upon well-studied hardness assumptions. Unfortunately, one of these assumptions is broken in quantum polynomial time. Luckily, the line work of Brakerski et al. Eurocrypt 2020, Gay-Pass STOC 2021, Wichs-Wee Eurocrypt 2021, Brakerski et al. ePrint 2021, Devadas et al. TCC 2021 simultaneously created new pathways to construct iO with plausible post-quantum security from new assumptions, namely a new form of circular security of LWE in the presence of leakages. At the same time, effective cryptanalysis of this line of work has also begun to emerge (Hopkins et al. Crypto 2021). It is important to identify the simplest possible conjectures that yield post-quantum iO and can be understood through known cryptanalytic tools. In that spirit, and in light of the cryptanalysis of Hopkins et al., recently Devadas et al. gave an elegant construction of iO from a fully-specified and simple-to-state assumption along with a thorough initial cryptanalysis. Our work gives a polynomial-time distinguisher on their "final assumption" for their scheme. Our algorithm is extremely simple to describe: Solve a carefully designed linear system arising out of the assumption. The argument of correctness of our algorithm, however, is nontrivial. We also analyze the "T-sum" version of the same assumption described by Devadas et. al. and under a reasonable conjecture rule out the assumption for any value of T that implies iO.

A wiretap coding scheme for a pair of noisy channels $(\chB,\chE)$ enables Alice to reliably communicate a message to Bob by sending its encoding over $\chB$, while hiding the message from an adversary Eve who obtains the same encoding over $\chE$. A necessary condition for the feasibility of writeup coding is that $\chB$ is not a {\em degradation} of $\chE$, namely Eve cannot simulate Bob’s view. While insufficient in the information-theoretic setting, a recent work of Ishai, Korb, Lou, and Sahai (Crypto 2022) showed that the non-degradation condition {\em is} sufficient in the computational setting, assuming idealized flavors of obfuscation. The question of basing a similar feasibility result on standard cryptographic assumptions was left open, even in simple special cases. In this work, we settle the question for all discrete memoryless channels where the (common) input alphabet of $\chB$ and $\chE$ is {\em binary}, and with arbitrary finite output alphabet, under the standard assumptions that indistinguishability obfuscation and injective PRGs exist. In particular, this establishes the feasibility of computational wiretap coding when $\chB$ is a binary symmetric channel with crossover probability $p$ and $\chE$ is a binary erasure channel with erasure probability $e$, where $e>2p$. On the information-theoretic side, our result builds on a new polytope characterization of channel degradation for pairs of binary-input channels, which may be of independent interest.

A wiretap coding scheme (Wyner in Bell Syst Tech J 54(8):1355–1387, 1975) enables Alice to reliably communicate a message m to an honest Bob by sending an encoding c over a noisy channel $$\textsf{ChB}$$ ChB , while at the same time hiding m from Eve who receives c over another noisy channel $$\textsf{ChE}$$ ChE . Wiretap coding is clearly impossible when $$\textsf{ChB}$$ ChB is a degraded version of $$\textsf{ChE}$$ ChE , in the sense that the output of $$\textsf{ChB}$$ ChB can be simulated using only the output of $$\textsf{ChE}$$ ChE . A classic work of Csiszár and Korner (IEEE Trans Inf Theory 24(3):339–348, 1978) shows that the converse does not hold. This follows from their full characterization of the channel pairs $$(\textsf{ChB},\textsf{ChE})$$ ( ChB , ChE ) that enable information-theoretic wiretap coding. In this work, we show that in fact the converse does hold when considering computational security ; that is, wiretap coding against a computationally bounded Eve is possible if and only if $$\textsf{ChB}$$ ChB is not a degraded version of $$\textsf{ChE}$$ ChE . Our construction assumes the existence of virtual black-box obfuscation of specific classes of “evasive” functions that generalize fuzzy point functions and can be heuristically instantiated using indistinguishability obfuscation. Finally, our solution has the appealing feature of being universal in the sense that Alice’s algorithm depends only on $$\textsf{ChB}$$ ChB and not on $$\textsf{ChE}$$ ChE .

A wiretap coding scheme (Wyner, Bell Syst.\ Tech.\ J.\ 1975) enables Alice to reliably communicate a message m to an honest Bob by sending an encoding c over a noisy channel ChB while at the same time hiding m from Eve who receives c over another noisy channel ChE. Wiretap coding is clearly impossible when ChB is a degraded version of ChE, in the sense that the output of ChB can be simulated using only the output of ChE. A classic work of Csiszár and Korner (IEEE Trans.\ Inf.\ Theory, 1978) shows that the converse does not hold. This follows from their full characterization of the channel pairs (ChB, ChE) that enable information-theoretic wiretap coding. In this work, we show that in fact the converse does hold when considering computational security; that is, wiretap coding against a computationally bounded Eve is possible if and only if ChB is not a degraded version of ChE. Our construction assumes the existence of virtual black-box (VBB) obfuscation of specific classes of ``evasive'' functions that generalize fuzzy point functions, and can be heuristically instantiated using indistinguishability obfuscation. Finally, our solution has the appealing feature of being universal in the sense that Alice's algorithm depends only on ChB and not on ChE.

All existing works building non-interactive zero-knowledge (NIZK) arguments for NP from the Learning With Errors (LWE) assumption have studied instantiating the Fiat-Shamir paradigm on a parallel repetition of an underlying honest-verifier zero knowledge (HVZK) sigma protocol, via an appropriately built correlation-intractable (CI) hash function from LWE. This technique has inherent efficiency losses that arise from parallel repetition. In this work, we show how to make use of the more efficient ``MPC in the Head'' technique for building an underlying honest-verifier protocol upon which to apply the Fiat-Shamir paradigm. To make this possible, we provide a new and more efficient construction of CI hash functions from LWE, using efficient algorithms for polynomial reconstruction as the main technical tool. We stress that our work provides a new and more efficient ``base construction'' for building LWE-based NIZK arguments for NP. Our protocol can be the building block around which other efficiency-focused bootstrapping techniques can be applied, such as the bootstrapping technique of Gentry et al. (Journal of Cryptology 2015).