We revisit the question of whether it is possible to build succinct non-interactive arguments (SNARGs) for all of NP under standard assumptions using non-signaling probabilistically checkable proofs [Kalai-Raz-Rothblum, STOC’ 14]. In particular, we observe that using exponential-length PCPs appears to circumvent all of the existing barriers.

We study several problems in the intersection of cryptography and complexity theory based on the following highlevel thesis.1)Obfuscation can serve as a general-purpose worst-case to average-case reduction, reducing the existence of various forms of cryptography to corresponding worst-case assumptions.2)We can therefore hope to overcome barriers in cryptography and average-case complexity by (i) making worstcase hardness assumptions beyond $P \neq N P$, and (ii) leveraging worst-case hardness reductions, either proved by traditional complexity-theoretic methods or facilitated further by cryptography.Concretely, our results include:•Optimal Hardness. Assuming sub-exponential indistinguishability obfuscation, we give fine-grained worst-case to average case reductions for circuit-SAT. In particular, if finding an NP-witness requires nearly brute-force time in the worst case, then the same is true for some efficiently sampleable distribution. In fact, we show that under these assumptions, there exist families of one-way functions with optimal time-probability security tradeoffs. Under an additional, stronger assumption - the optimal non-deterministic hardness of refuting circuit-SAT - we construct additional cryptographic primitives such as PRGs and public-key encryption that have such optimal timeadvantage security tradeoffs.•Direct Product Hardness. Again assuming iO and optimal non-deterministic hardness of SAT refutation, we show that the “(search) k-fold SAT problem” - the computational task of finding satisfying assignments to k circuit-SAT instances simultaneously - has (optimal) hardness roughly $\left(T / 2^{n}\right)^{k}$ for time T algorithms. In fact, we build “optimally secure one-way product functions” (Holmgren-Lombardi, FOCS ‘18), demonstrating that optimal direct product theorems hold for some choice of one-way function family.•Single-Input Correlation Intractability. Assuming either iO or LWE, we show a worst-case to average-case reduction for strong forms of single-input correlation intractability. That is, powerful forms of correlation-intractable hash functions exist provided that a collection of worst-case “correlationfinding” problems are hard.•Non-interactive Proof of Quantumness. Assuming subexponential iO and OWFs, we give a non-interactive proof of quantumness based on the worst-case hardness of the whitebox Simon problem. In particular, this proof of quantumness result does not explicitly assume quantum advantage for an average-case task.To help prove our first two results, we show along the way how to improve the Goldwasser-Sipser “set lower bound” protocol to have communication complexity quadratically smaller in the multiplicative approximation error $\varepsilon$.

STOC ’25, Prague, Czechia

We construct a succinct non-interactive argument (SNARG) system for every NP language L that has a propositional proof of non-membership, i.e. of x∉ L. The soundness of our SNARG system relies on the hardness of the learning with errors (LWE) problem. The common reference string (CRS) in our construction grows with the space required to verify the propositional proof, and the size of the proof grows poly-logarithmically in the length of the propositional proof. Unlike most of the literature on SNARGs, our result implies SNARGs for languages L with proof length shorter than logarithmic in the deterministic time complexity of L. Our SNARG improves over prior SNARGs for such “hard” NP languages (Sahai and Waters, STOC 2014, Jain and Jin, FOCS 2022) in several ways: 1) For languages with polynomial-length propositional proofs of non-membership, our SNARGs are based on a single, polynomial-time falsifiable assumption, namely LWE. 2) Our construction handles super-polynomial length propositional proofs, as long as they have bounded space, under the subexponential LWE assumption. 3) Our SNARGs have a transparent setup, meaning that no private randomness is required to generate the CRS. Moreover, our approach departs dramatically from these prior works: we show how to design SNARGs for hard languages without publishing a program (in the CRS) that has the power to verify NP witnesses. The key new idea in our construction is what we call a “locally unsatisfiable extension” of the NP verification circuit {Cx}x. We say that an NP verifier has a locally unsatisfiable extension if for every x∉L, there exists an extension Ex of Cx that is not even locally satisfiable in the sense of a local assignment generator [Paneth-Rothblum, TCC 2017]. Crucially, we allow Ex to be depend arbitrarily on x rather than being efficiently constructible. In this work, we show – via a “hash-and-BARG” for a hidden, encrypted computation – how to build SNARGs for all languages with locally unsatisfiable extensions. We additionally show that propositional proofs of unsatisfiability generically imply the existence of locally unsatisfiable extensions, which allows us to deduce our main results. As an illustrative example, our results imply a SNARG for the decisional Diffie-Hellman (DDH) language under the LWE assumption.

The Unitary Synthesis Problem (Aaronson-Kuperberg 2007) asks whether any n-qubit unitary U can be implemented by an efficient quantum algorithm A augmented with an oracle that computes an arbitrary Boolean function f. In other words, can the task of implementing any unitary be efficiently reduced to the task of implementing any Boolean function? In this work, we prove a one-query lower bound for unitary synthesis. We show that there exist unitaries U such that no quantum polynomial-time oracle algorithm Af can implement U, even approximately, if it only makes one (quantum) query to f. Our approach also has implications for quantum cryptography: we prove (relative to a random oracle) the existence of quantum cryptographic primitives that remain secure against all one-query adversaries Af. Since such one-query algorithms can decide any language, solve any classical search problem, and even prepare any quantum state, our result suggests that implementing random unitaries and breaking quantum cryptography may be harder than all of these tasks. To prove this result, we formulate unitary synthesis as an efficient challenger-adversary game, which enables proving lower bounds by analyzing the maximum success probability of an adversary Af. Our main technical insight is to identify a natural spectral relaxation of the one-query optimization problem, which we bound using tools from random matrix theory. We view our framework as a potential avenue to rule out polynomial-query unitary synthesis, and we state conjectures in this direction.

We show a general method of compiling any k-prover non-local game into a single-prover (computationally sound) interactive game maintaining the same quantum completeness and classical soundness guarantees, up to a negligible additive factor in a security parameter. Our compiler uses any quantum homomorphic encryption scheme (Mahadev, FOCS 2018; Brakerski, CRYPTO 2018) satisfying a natural form of correctness with respect to auxiliary quantum input. The homomorphic encryption scheme is used as a cryptographic mechanism to simulate the effect of spatial separation, and is required to evaluate k−1 prover strategies out of k on encrypted queries.

We show how to generically improve the succinctness of non-interactive publicly verifiable batch argument (BARG) systems. In particular, we show (under a mild additional assumption) how to convert a BARG that generates proofs of length poly (m)· k1−є, where m is the length of a single instance and k is the number of instances being batched, into one that generates proofs of length poly (m, logk), which is the gold standard for succinctness of BARGs. By prior work, such BARGs imply the existence of SNARGs for deterministic time T computation with succinctness poly(logT).

When do classical zero-knowledge protocols remain secure against quantum attacks? In this work, we develop the techniques, tools, and abstractions necessary to answer this question for foundational protocols:1)We prove that the Goldreich-Micali-Wigderson protocol for graph non-isomorphism and the Feige-Shamir protocol for NP remain zero-knowledge against quantum adversaries. At the heart of our proof is a new quantum rewinding technique that enables extracting information from multiple invocations of a quantum adversary without disturbing its state.2)We prove that the Goldreich-Kahan protocol for NP is post-quantum zero knowledge using a simulator that can be seen as a natural quantum extension of the classical simulator.Our results achieve negligible simulation error, appearing to contradict a recent impossibility result due to Chia-Chung-Liu-Yamakawa (FOCS 2021). This brings us to our final contribution:3.We introduce coherent-runtime expected quantum polynomial time, a simulation notion that (a) precisely captures all of our zero-knowledge simulators, (b) cannot break any polynomial hardness assumptions, (c) implies strict polynomial-time ε-simulation and (d) is not subject to the CCLY impossibility. In light of our positive results and the CCLY negative results, we propose coherent-runtime simulation to be the appropriate quantum analogue of classical expected polynomial-time simulation.

A hash function family ℋ is correlation intractable for a t-input relation ℛ if, given a random function h chosen from ℋ, it is hard to find x_1,…,x_t such that ℛ(x_1,…,x_t,h(x₁),…,h(x_t)) is true. Among other applications, such hash functions are a crucial tool for instantiating the Fiat-Shamir heuristic in the plain model, including the only known NIZK for NP based on the learning with errors (LWE) problem (Peikert and Shiehian, CRYPTO 2019). We give a conceptually simple and generic construction of single-input CI hash functions from shift-hiding shiftable functions (Peikert and Shiehian, PKC 2018) satisfying an additional one-wayness property. This results in a clean abstract framework for instantiating CI, and also shows that a previously existing function family (PKC 2018) was already CI under the LWE assumption. In addition, our framework transparently generalizes to other settings, yielding new results: - We show how to instantiate certain forms of multi-input CI under the LWE assumption. Prior constructions either relied on a very strong "brute-force-is-best" type of hardness assumption (Holmgren and Lombardi, FOCS 2018) or were restricted to "output-only" relations (Zhandry, CRYPTO 2016). - We construct single-input CI hash functions from indistinguishability obfuscation (iO) and one-way permutations. Prior constructions relied essentially on variants of fully homomorphic encryption that are impossible to construct from such primitives. This result also generalizes to more expressive variants of multi-input CI under iO and additional standard assumptions.

In a seminal work, Goldreich, Micali and Wigderson (CRYPTO ’86) demonstrated the wide applicability of zero-knowledge proofs by constructing such a proof system for the NP-complete problem of graph 3-coloring. A long-standing open question has been whether parallel repetition of their protocol preserves zero knowledge. In this work, we answer this question in the negative, assuming a standard cryptographic assumption (i.e., the hardness of learning with errors (LWE)).

We give new instantiations of the Fiat-Shamir transform using explicit, efficiently computable hash functions. We improve over prior work by reducing the security of these protocols to qualitatively simpler and weaker computational hardness assumptions. As a consequence of our framework, we obtain the following concrete results.

Constructing collision-resistant hash families (CRHFs) from one-way functions is a long-standing open problem and source of frustration in theoretical cryptography. In fact, there are strong negative results: black-box separations from one-way functions that are 2-(1-0(1))n-secure against polynomial time adversaries (Simon, EUROCRYPT '98) and even from indistinguishability obfuscation (Asharov and Segev, FOCS '15). In this work, we formulate a mild strengthening of exponentially secure one-way functions, and we construct CRHFs from such functions. Specifically, our security notion requires that every polynomial time algorithm has at most 2-n· negl(n) probability of inverting two independent challenges. More generally, we consider the problem of simultaneously inverting k functions f1,.. . , fk, which we say constitute a “one-way product function” (OWPF). We show that sufficiently hard OWPFs yield hash families that are multi-input correlation intractable (Canetti, Goldreich, and Halevi, STOC '98) with respect to all sparse (bounded arity) output relations. Additionally assuming indistinguishability obfuscation, we construct hash families that achieve a broader notion of correlation intractability, extending the recent work of Kalai, Rothblum, and Rothblum (CRYPTO '17). In particular, these families are sufficient to instantiate the Fiat-Shamir heuristic in the plain model for a natural class of interactive proofs. An interesting consequence of our results is a potential new avenue for bypassing black-box separations. In particular, proving (with necessarily non-black-box techniques) that parallel repetition amplifies the hardness of specific one-way functions - for example, all oneway permutations - suffices to directly bypass Simon's impossibility result.