We provide a zero-knowledge argument for arithmetic circuit satisfiability with a communication complexity that grows logarithmically in the size of the circuit. The round complexity is also logarithmic and for an arithmetic circuit with fan-in 2 gates the computation of the prover and verifier is linear in the size of the circuit. The soundness of our argument relies solely on the well-established discrete logarithm assumption in prime order groups. At the heart of our new argument system is an efficient zero-knowledge argument of knowledge of openings of two Pedersen multicommitments satisfying an inner product relation, which is of independent interest. The inner product argument requires logarithmic communication, logarithmic interaction and linear computation for both the prover and the verifier. We also develop a scheme to commit to a polynomial and later reveal the evaluation at an arbitrary point, in a verifiable manner. This is used to build an optimized version of the constant round square root complexity argument of Groth (CRYPTO 2009), which reduces both communication and round complexity.Informally, a zero-knowledge argument involves two parties, the prover and the verifier, and allows the prover to prove to the verifier that a particular statement is true, without revealing anything else about the statement itself. Statements are of the form u ∈ L, where L is a language in NP. We call w a witness for a statement u if (u, w) ∈ R, where R is a polynomial time decidable binary relation associated with L. We require the zero-knowledge argument to be complete, sound and zero-knowledge. Completeness:A prover with a witness w for u ∈ L can convince the verifier of this fact. Soundness: A prover cannot convince a verifier when u / ∈ L. Zero-knowledge: The interaction should not reveal anything to the verifier except that u ∈ L. In particular, it should not reveal the prover's witness w.Our goal is to build an efficient argument system for the satisfiability of an arithmetic circuit, i.e., a circuit that consists of addition and multiplication gates over a finite field Z p . Moreover we want to base the security of this argument solely on the discrete logarithm assumption: this will provide both strong security guarantees and good efficiency since there exists no known attacks better than generic ones for well-chosen elliptic curve subgroups.The most efficient zero-knowledge arguments solely based on the discrete logarithm assumption are Groth's protocol based on linear algebra [Gro09b] and its variant by Seo [Seo11]. Both of these protocols have a communication complexity that is proportional to the square root of the circuit size. This square root complexity has since then appeared as a (perhaps fundamental) barrier for discrete logarithm-based arguments for circuit satisfiability. Our ContributionsWe provide an honest verifier zero-knowledge argument for arithmetic circuit satisfiability based on the discrete logarithm assumption that only requires a logarithmic communication complexity. Our argument has ...

Let O be a maximal order in a definite quaternion algebra over Q of prime discriminant p, and a small prime. We describe a probabilistic algorithm which, for a given left O-ideal, computes a representative in its left ideal class of -power norm. In practice the algorithm is efficient and, subject to heuristics on expected distributions of primes, runs in expected polynomial time. This solves the underlying problem for a quaternion analog of the Charles-Goren-Lauter hash function, and has security implications for the original CGL construction in terms of supersingular elliptic curves.

Abstract. In the last two decades, many computational problems arising in cryptography have been successfully reduced to various systems of polynomial equations. In this paper, we revisit a class of polynomial systems introduced by Faugère, Perret, Petit and Renault. Based on new experimental results and heuristic evidence, we conjecture that their degrees of regularity are only slightly larger than the original degrees of the equations, resulting in a very low complexity compared to generic systems. We then revisit the application of these systems to the elliptic curve discrete logarithm problem (ECDLP) for binary curves. Our heuristic analysis suggests that an index calculus variant due to Diem requires a subexponential number of bit operations O(2 c n 2/3 log n ) over the binary field F2n , where c is a constant smaller than 2. According to our estimations, generic discrete logarithm methods are outperformed for any n > N where N ≈ 2000, but elliptic curves of currently recommended key sizes (n ≈ 160) are not immediately threatened. The analysis can be easily generalized to other extension fields.

We study cryptosystems based on supersingular isogenies. This is an active area of research in post-quantum cryptography. Our first contribution is to give a very powerful active attack on the supersingular isogeny encryption scheme. This attack can only be prevented by using a (relatively expensive) countermeasure. Our second contribution is to show that the security of all schemes of this type depends on the difficulty of computing the endomorphism ring of a supersingular elliptic curve. This result gives significant insight into the difficulty of the isogeny problem that underlies the security of these schemes. Our third contribution is to give a reduction that uses partial knowledge of shared keys to determine an entire shared key. This can be used to retrieve the secret key, given information leaked from a side-channel attack on the key exchange protocol. A corollary of this work is the first bit security result for the supersingular isogeny key exchange: Computing any component of the j-invariant is as hard as computing the whole j-invariant. Our paper therefore provides an improved understanding of the security of these cryptosystems. We stress that our work does not imply that these systems are insecure, or that they should not be used. However, it highlights that implementations of these schemes will need to take account of the risks associated with various active and side-channel attacks.

Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document. When citing, please reference the published version. Take down policy While the University of Birmingham exercises care and attention in making items available there are rare occasions when an item has been uploaded in error or has been deemed to be commercially or otherwise sensitive.