International Association for Cryptologic Research

International Association
for Cryptologic Research


Alexander R. Block


Field-Agnostic SNARKs from Expand-Accumulate Codes
Efficient realizations of succinct non-interactive arguments of knowledge (SNARK) have gained popularity due to their practical applications in various domains. Among existing schemes, those based on error-correcting codes are of particular interest because of their good concrete efficiency, transparent setup, and plausible post-quantum security. However, many existing code-based SNARKs suffer from the disadvantage that they only work over specific finite fields. In this work, we construct a code-based SNARK that does not rely on any specific underlying field; i.e., it is \emph{field-agnostic}. Our construction follows the framework of Brakedown and builds a polynomial commitment scheme (and hence a SNARK) based on recently introduced \emph{expand-accumulate codes}. Our work generalizes these codes to arbitrary finite fields; our main technical contribution is showing that, with high probability, these codes have constant rate and constant relative distance (crucial properties for building efficient SNARKs), solving an open problem from prior work. As a result of our work we obtain a SNARK where, for a statement of size $M$, the prover time is $O(M\log M)$ and the proof size is $O(\sqrt{M})$. We demonstrate the concrete efficiency of our scheme empirically via experiments. Proving ECDSA verification on the secp256k1 curve requires only 0.23s for proof generation, 2~orders of magnitude faster than SNARKs that are not field-agnostic. Compared to the original Brakedown result (which is also field-agnostic), we obtain proofs that are 1.9--2.8$\times$ smaller due to the good concrete distance of our underlying error-correcting code, while introducing only a small overhead of 1.2$\times$ in the prover time.
Fiat-Shamir Security of FRI and Related SNARKs
We establish new results on the Fiat-Shamir (FS) security of several protocols that are widely used in practice, and we provide general tools for establishing similar results for others. More precisely, we: (1) prove the FS security of the FRI and batched FRI protocols; (2) analyze a general class of protocols, which we call \emph{$\delta$-correlated}, that use low-degree proximity testing as a subroutine (this includes many ``Plonk-like'' protocols (e.g., Plonky2 and Redshift), ethSTARK, RISC Zero, etc.); and (3) prove FS security of the aforementioned ``Plonk-like'' protocols, and sketch how to prove the same for the others. We obtain our first result by analyzing the round-by-round (RBR) soundness and RBR knowledge soundness of FRI. For the second result, we prove that if a $\delta$-correlated protocol is RBR (knowledge) sound under the assumption that adversaries always send low-degree polynomials, then it is RBR (knowledge) sound in general. Equipped with this tool, we prove our third result by formally showing that ``Plonk-like'' protocols are RBR (knowledge) sound under the assumption that adversaries always send low-degree polynomials. We then outline analogous arguments for the remainder of the aforementioned protocols. To the best of our knowledge, ours is the first formal analysis of the Fiat-Shamir security of FRI and widely deployed protocols that invoke it.
Time- and Space-Efficient Arguments from Groups of Unknown Order 📺
We construct public-coin time- and space-efficient zero-knowledge arguments for NP. For every time T and space S non-deterministic RAM computation, the prover runs in time T * polylog(T) and space S * polylog(T), and the verifier runs in time n * polylog(T), where n is the input length. Our protocol relies on hidden order groups, which can be instantiated with a trusted setup from the hardness of factoring (products of safe primes), or without a trusted setup using class groups. The argument-system can heuristically be made non-interactive using the Fiat-Shamir transform. Our proof builds on DARK (Bunz et al., Eurocrypt 2020), a recent succinct and efficiently verifiable polynomial commitment scheme. We show how to implement a variant of DARK in a time- and space-efficient way. Along the way we: 1. Identify a significant gap in the proof of security of Dark. 2. Give a non-trivial modification of the DARK scheme that overcomes the aforementioned gap. The modified version also relies on significantly weaker cryptographic assumptions than those in the original DARK scheme. Our proof utilizes ideas from the theory of integer lattices in a novel way. 3. Generalize Pietrzak's (ITCS 2019) proof of exponentiation (PoE) protocol to work with general groups of unknown order (without relying on any cryptographic assumption). In proving these results, we develop general-purpose techniques for working with (hidden order) groups, which may be of independent interest.
Public-Coin Zero-Knowledge Arguments with (almost) Minimal Time and Space Overheads 📺
Zero-knowledge protocols enable the truth of a mathematical statement to be certified by a verifier without revealing any other information. Such protocols are a cornerstone of modern cryptography and recently are becoming more and more practical. However, a major bottleneck in deployment is the efficiency of the prover and, in particular, the space-efficiency of the protocol. For every $\mathsf{NP}$ relation that can be verified in time $T$ and space $S$, we construct a public-coin zero-knowledge argument in which the prover runs in time $T \cdot \mathrm{polylog}(T)$ and space $S \cdot \mathrm{polylog}(T)$. Our proofs have length $\mathrm{polylog}(T)$ and the verifier runs in time $T \cdot \mathrm{polylog}(T)$ (and space $\mathrm{polylog}(T)$). Our scheme is in the random oracle model and relies on the hardness of discrete log in prime-order groups. Our main technical contribution is a new space efficient \emph{polynomial commitment scheme} for multi-linear polynomials. Recall that in such a scheme, a sender commits to a given multi-linear polynomial $P:\mathbb{F}^n \to \mathbb{F}$ so that later on it can prove to a receiver statements of the form ``$P(x)=y$''. In our scheme, which builds on commitments schemes of Bootle et al. (Eurocrypt 2016) and B{\"u}nz et al. (S\&P 2018), we assume that the sender is given multi-pass streaming access to the evaluations of $P$ on the Boolean hypercube and we show how to implement both the sender and receiver in roughly time $2^n$ and space $n$ and with communication complexity roughly $n$.
Secure Computation Using Leaky Correlations (Asymptotically Optimal Constructions)
Most secure computation protocols can be effortlessly adapted to offload a significant fraction of their computationally and cryptographically expensive components to an offline phase so that the parties can run a fast online phase and perform their intended computation securely. During this offline phase, parties generate private shares of a sample generated from a particular joint distribution, referred to as the correlation. These shares, however, are susceptible to leakage attacks by adversarial parties, which can compromise the security of the secure computation protocol. The objective, therefore, is to preserve the security of the honest party despite the leakage performed by the adversary on her share.Prior solutions, starting with n-bit leaky shares, either used 4 messages or enabled the secure computation of only sub-linear size circuits. Our work presents the first 2-message secure computation protocol for 2-party functionalities that have $$\varTheta (n)$$ circuit-size despite $$\varTheta (n)$$-bits of leakage, a qualitatively optimal result. We compose a suitable 2-message secure computation protocol in parallel with our new 2-message correlation extractor. Correlation extractors, introduced by Ishai, Kushilevitz, Ostrovsky, and Sahai (FOCS–2009) as a natural generalization of privacy amplification and randomness extraction, recover “fresh” correlations from the leaky ones, which are subsequently used by other cryptographic protocols. We construct the first 2-message correlation extractor that produces $$\varTheta (n)$$-bit fresh correlations even after $$\varTheta (n)$$-bit leakage.Our principal technical contribution, which is of potential independent interest, is the construction of a family of multiplication-friendly linear secret sharing schemes that is simultaneously a family of small-bias distributions. We construct this family by randomly “twisting then permuting” appropriate Algebraic Geometry codes over constant-size fields.