International Association
for Cryptologic Research

Interactive oracle proofs (IOPs) are a multi-round generalization of probabilistically checkable proofs that play a fundamental role in the construction of efficient cryptographic proofs. We present an IOP that simultaneously achieves the properties of zero knowledge, linear-time proving, and polylogarithmic-time verification. We construct a zero-knowledge IOP where, for the satisfiability of an $N$-gate arithmetic circuit over any field of size $\Omega(N)$, the prover uses $O(N)$ field operations and the verifier uses $\polylog(N)$ field operations (with proof length $O(N)$ and query complexity $\polylog(N)$). Polylogarithmic verification is achieved in the holographic setting for every circuit (the verifier has oracle access to a linear-time-computable encoding of the circuit whose satisfiability is being proved). Our result implies progress on a basic goal in the area of efficient zero knowledge. Via a known transformation, we obtain a zero knowledge argument system where the prover runs in linear time and the verifier runs in polylogarithmic time; the construction is plausibly post-quantum and only makes a black-box use of lightweight cryptography (collision-resistant hash functions).

We introduce a new class of succinct arguments, that we call elastic. Elastic SNARKs allow the prover to allocate different resources (such as memory and time) depending on the execution environment and the statement to prove. The resulting output is independent of the prover’s configuration. To study elastic SNARKs, we extend the streaming paradigm of [Block et al., TCC’20]. We provide a definitional framework for elastic polynomial interactive oracle proofs for R1CS instances and design a compiler which transforms an elastic PIOP into a preprocessing argument system that supports streaming or random access to its inputs. Depending on the configuration, the prover will choose different trade-offs for time (either linear, or quasilinear) and memory (either linear, or logarithmic). We prove the existence of elastic SNARKS by presenting Gemini, a novel FFT-free preprocessing argument. We prove its security and develop a proof-of-concept implementation in Rust based on the arkworks framework. We provide benchmarks for large R1CS instances of tens of billions of gates on a single machine.

We introduce a class of interactive protocols, which we call *sumcheck arguments*, that establishes a novel connection between the sumcheck protocol (Lund et al. JACM 1992) and folding techniques for Pedersen commitments (Bootle et al. EUROCRYPT 2016). Informally, we consider a general notion of bilinear commitment over modules, and show that the sumcheck protocol applied to a certain polynomial associated with the commitment scheme yields a succinct argument of knowledge for openings of the commitment. Building on this, we additionally obtain succinct arguments for the NP-complete language R1CS over certain rings. Sumcheck arguments enable us to recover as a special case numerous prior works in disparate cryptographic settings (such as discrete logarithms, pairings, RSA groups, lattices), providing one abstract framework to understand them all. Further, we answer open questions raised in prior works, such as obtaining a lattice-based succinct argument from the SIS assumption for satisfiability problems over rings.

Today's most compact zero-knowledge arguments are based on the hardness of the discrete logarithm problem and related classical assumptions. If one is interested in quantum-safe solutions, then all of the known techniques stem from the PCP-based framework of Kilian (STOC 92) which can be instantiated based on the hardness of any collision-resistant hash function. Both approaches produce asymptotically logarithmic sized arguments but, by exploiting extra algebraic structure, the discrete logarithm arguments are a few orders of magnitude more compact in practice than the generic constructions.\\ In this work, we present the first (poly)-logarithmic \emph{post-quantum} zero-knowledge arguments that deviate from the PCP approach. At the core of succinct zero-knowledge proofs are succinct commitment schemes (in which the commitment and the opening proof are sub-linear in the message size), and we propose two such constructions based on the hardness of the (Ring)-Short Integer Solution (Ring-SIS) problem, each having certain trade-offs. For commitments to $N$ secret values, the communication complexity of our first scheme is $\tilde{O}(N^{1/c})$ for any positive integer $c$, and $O(\log^2 N)$ for the second. %Both of our protocols have somewhat large \emph{slack}, which in lattice constructions is the ratio of the norm of the extracted secrets to the norm of the secrets that the honest prover uses in the proof. The lower this factor, the smaller we can choose the practical parameters. For a fixed value of this factor, our $\tilde{O}(N^{1/c})$-argument actually achieves lower communication complexity. Both of these are a significant theoretical improvement over the previously best lattice construction by Bootle et al. (CRYPTO 2018) which gave $O(\sqrt{N})$-sized proofs.

Minimizing the computational cost of the prover is a central goal in the area of succinct arguments. In particular, it remains a challenging open problem to construct a succinct argument where the prover runs in linear time and the verifier runs in polylogarithmic time. We make progress towards this goal by presenting a new linear-time probabilistic proof. For any fixed ? > 0, we construct an interactive oracle proof (IOP) that, when used for the satisfiability of an N-gate arithmetic circuit, has a prover that uses O(N) field operations and a verifier that uses O(N^?) field operations. The sublinear verifier time is achieved in the holographic setting for every circuit (the verifier has oracle access to a linear-size encoding of the circuit that is computable in linear time). When combined with a linear-time collision-resistant hash function, our IOP immediately leads to an argument system where the prover performs O(N) field operations and hash computations, and the verifier performs O(N^?) field operations and hash computations (given a short digest of the N-gate circuit).

Group signatures allow members of a group to anonymously sign on behalf of the group. Membership is administered by a designated group manager. The group manager can also reveal the identity of a signer if and when needed to enforce accountability and deter abuse. For group signatures to be applicable in practice, they need to support fully dynamic groups, i.e., users may join and leave at any time. Existing security definitions for fully dynamic group signatures are informal, have shortcomings, and are mutually incompatible. We fill the gap by providing a formal rigorous security model for fully dynamic group signatures. Our model is general and is not tailored toward a specific design paradigm and can therefore, as we show, be used to argue about the security of different existing constructions following different design paradigms. Our definitions are stringent and when possible incorporate protection against maliciously chosen keys. We consider both the case where the group management and tracing signatures are administered by the same authority, i.e., a single group manager, and also the case where those roles are administered by two separate authorities, i.e., a group manager and an opening authority. We also show that a specialization of our model captures existing models for static and partially dynamic schemes. In the process, we identify a subtle gap in the security achieved by group signatures using revocation lists. We show that in such schemes new members achieve a slightly weaker notion of traceability. The flexibility of our security model allows to capture such relaxation of traceability.

A key component of many lattice-based protocols is a zero-knowledge proof of knowledge of a vector $$\vec {s}$$ with small coefficients satisfying $$A\vec {s}=\vec {u}\bmod \,q$$ . While there exist fairly efficient proofs for a relaxed version of this equation which prove the knowledge of $$\vec {s}'$$ and c satisfying $$A\vec {s}'=\vec {u}c$$ where $$\Vert \vec {s}'\Vert \gg \Vert \vec {s}\Vert $$ and c is some small element in the ring over which the proof is performed, the proofs for the exact version of the equation are considerably less practical. The best such proof technique is an adaptation of Stern’s protocol (Crypto ’93), for proving knowledge of nearby codewords, to larger moduli. The scheme is a $$\varSigma $$ -protocol, each of whose iterations has soundness error $$2{/}3$$ , and thus requires over 200 repetitions to obtain soundness error of $$2^{-128}$$ , which is the main culprit behind the large size of the proofs produced. In this paper, we propose the first lattice-based proof system that significantly outperforms Stern-type proofs for proving knowledge of a short $$\vec {s}$$ satisfying $$A\vec {s}=\vec {u}\bmod \,q$$ . Unlike Stern’s proof, which is combinatorial in nature, our proof is more algebraic and uses various relaxed zero-knowledge proofs as sub-routines. The main savings in our proof system comes from the fact that each round has soundness error of $$1{/}n$$ , where n is the number of columns of A. For typical applications, n is a few thousand, and therefore our proof needs to be repeated around 10 times to achieve a soundness error of $$2^{-128}$$ . For concrete parameters, it produces proofs that are around an order of magnitude smaller than those produced using Stern’s approach.

We propose the first zero-knowledge argument with sub-linear communication complexity for arithmetic circuit satisfiability over a prime $${p}$$ whose security is based on the hardness of the short integer solution (SIS) problem. For a circuit with $${N}$$ gates, the communication complexity of our protocol is $$O\left( \sqrt{{N}{\lambda }\log ^3{{N}}}\right) $$ , where $${\lambda }$$ is the security parameter. A key component of our construction is a surprisingly simple zero-knowledge proof for pre-images of linear relations whose amortized communication complexity depends only logarithmically on the number of relations being proved. This latter protocol is a substantial improvement, both theoretically and in practice, over the previous results in this line of research of Damgård et al. (CRYPTO 2012), Baum et al. (CRYPTO 2016), Cramer et al. (EUROCRYPT 2017) and del Pino and Lyubashevsky (CRYPTO 2017), and we believe it to be of independent interest.

Bootle et al. (EUROCRYPT 2016) construct an extremely efficient zero-knowledge argument for arithmetic circuit satisfiability in the discrete logarithm setting. However, the argument does not treat relations involving commitments, and furthermore, for simple polynomial relations, the complex machinery employed is unnecessary.In this work, we give a framework for expressing simple relations between commitments and field elements, and present a zero-knowledge argument which, by contrast with Bootle et al., is constant-round and uses fewer group operations, in the case where the polynomials in the relation have low degree. Our method also directly yields a batch protocol, which allows many copies of the same relation to be proved and verified in a single argument more efficiently with only a square-root communication overhead in the number of copies.We instantiate our protocol with concrete polynomial relations to construct zero-knowledge arguments for membership proofs, polynomial evaluation proofs, and range proofs. Our work can be seen as a unified explanation of the underlying ideas of these protocols. In the instantiations of membership proofs and polynomial evaluation proofs, we also achieve better efficiency than the state of the art.

This paper is devoted to analyzing the variant of Regev’s learning with errors (LWE) problem in which modular reduction is omitted: namely, the problem (ILWE) of recovering a vector $$\mathbf {s}\in \mathbb {Z}^n$$ given polynomially many samples of the form $$(\mathbf {a},\langle \mathbf {a},\mathbf {s}\rangle + e)\in \mathbb {Z}^{n+1}$$ where $$\mathbf { a}$$ and e follow fixed distributions. Unsurprisingly, this problem is much easier than LWE: under mild conditions on the distributions, we show that the problem can be solved efficiently as long as the variance of e is not superpolynomially larger than that of $$\mathbf { a}$$. We also provide almost tight bounds on the number of samples needed to recover $$\mathbf {s}$$.Our interest in studying this problem stems from the side-channel attack against the BLISS lattice-based signature scheme described by Espitau et al. at CCS 2017. The attack targets a quadratic function of the secret that leaks in the rejection sampling step of BLISS. The same part of the algorithm also suffers from a linear leakage, but the authors claimed that this leakage could not be exploited due to signature compression: the linear system arising from it turns out to be noisy, and hence key recovery amounts to solving a high-dimensional problem analogous to LWE, which seemed infeasible. However, this noisy linear algebra problem does not involve any modular reduction: it is essentially an instance of ILWE, and can therefore be solved efficiently using our techniques. This allows us to obtain an improved side-channel attack on BLISS, which applies to 100% of secret keys (as opposed to $${\approx }7\%$$ in the CCS paper), and is also considerably faster.

There have been tremendous advances in reducing interaction, communication and verification time in zero-knowledge proofs but it remains an important challenge to make the prover efficient. We construct the first zero-knowledge proof of knowledge for the correct execution of a program on public and private inputs where the prover computation is nearly linear time. This saves a polylogarithmic factor in asymptotic performance compared to current state of the art proof systems.We use the TinyRAM model to capture general purpose processor computation. An instance consists of a TinyRAM program and public inputs. The witness consists of additional private inputs to the program. The prover can use our proof system to convince the verifier that the program terminates with the intended answer within given time and memory bounds. Our proof system has perfect completeness, statistical special honest verifier zero-knowledge, and computational knowledge soundness assuming linear-time computable collision-resistant hash functions exist. The main advantage of our new proof system is asymptotically efficient prover computation. The prover’s running time is only a superconstant factor larger than the program’s running time in an apples-to-apples comparison where the prover uses the same TinyRAM model. Our proof system is also efficient on the other performance parameters; the verifier’s running time and the communication are sublinear in the execution time of the program and we only use a log-logarithmic number of rounds.