Zero-Knowledge Argument for Polynomial Evaluation with Application to Blacklists

Verification of a polynomial’s evaluation in a secret committed value plays a role in cryptographic applications such as non-membership or membership proofs. We construct a novel special honest verifier zero-knowledge argument for correct polynomial evaluation. The argument has logarithmic communication cost in the degree of the polynomial, which is a significant improvement over the state of the art with cubic root complexity at best. The argument is relatively efficient to generate and very fast to verify compared to previous work. The argument has a simple public-coin 3-move structure and only relies on the discrete logarithm assumption. The polynomial evaluation argument can be used as a building block to construct zero-knowledge membership and non-membership arguments with communication that is logarithmic in the size of the blacklist. Non-membership proofs can be used to design anonymous blacklisting schemes allowing online services to block misbehaving users without learning the identity of the user. They also allow the blocking of single users of anonymization networks without blocking the whole network

On the efficiency of revocation in RSA-based anonymous systems

© 2016 IEEEThe problem of revocation in anonymous authentication systems is subtle and has motivated a lot of work. One of the preferable solutions consists in maintaining either a whitelist L-W of non-revoked users or a blacklist L-B of revoked users, and then requiring users to additionally prove, when authenticating themselves, that they are in L-W (membership proof) or that they are not in L-B (non-membership proof). Of course, these additional proofs must not break the anonymity properties of the system, so they must be zero-knowledge proofs, revealing nothing about the identity of the users. In this paper, we focus on the RSA-based setting, and we consider the case of non-membership proofs to blacklists L = L-B. The existing solutions for this setting rely on the use of universal dynamic accumulators; the underlying zero-knowledge proofs are bit complicated, and thus their efficiency; although being independent from the size of the blacklist L, seems to be improvable. Peng and Bao already tried to propose simpler and more efficient zero-knowledge proofs for this setting, but we prove in this paper that their protocol is not secure. We fix the problem by designing a new protocol, and formally proving its security properties. We then compare the efficiency of the new zero-knowledge non-membership protocol with that of the protocol, when they are integrated with anonymous authentication systems based on RSA (notably, the IBM product Idemix for anonymous credentials). We discuss for which values of the size k of the blacklist L, one protocol is preferable to the other one, and we propose different ways to combine and implement the two protocols.Postprint (author's final draft

Lattice-Based proof of a shuffle

In this paper we present the first fully post-quantum proof of a shuffle for RLWE encryption schemes. Shuffles are commonly used to construct mixing networks (mix-nets), a key element to ensure anonymity in many applications such as electronic voting systems. They should preserve anonymity even against an attack using quantum computers in order to guarantee long-term privacy. The proof presented in this paper is built over RLWE commitments which are perfectly binding and computationally hiding under the RLWE assumption, thus achieving security in a post-quantum scenario. Furthermore we provide a new definition for a secure mixing node (mix-node) and prove that our construction satisfies this definition.Peer ReviewedPostprint (author's final draft

Efficient Batch Zero-Knowledge Arguments for Low Degree Polynomials

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

Practical zero-knowledge Protocols based on the discrete logarithm Assumption

Zero-knowledge proofs were introduced by Goldwasser, Micali, and Rackoff. A zero-knowledge proof allows a prover to demonstrate knowledge of some information, for example that they know an element which is a member of a list or which is not a member of a list, without disclosing any further information about that element. Existing constructions of zero-knowledge proofs which can be applied to all languages in NP are impractical due to their communication and computational complexity. However, it has been known since Guillou and Quisquater's identification protocol from 1988 and Schnorr's identification protocol from 1991 that practical zero-knowledge protocols for specific problems exist. Because of this, a lot of work was undertaken over the recent decades to find practical zero-knowledge proofs for various other specific problems, and in recent years many protocols were published which have improved communication and computational complexity. Nevertheless, to find more problems which have an efficient and practical zero-knowledge proof system and which can be used as building blocks for other protocols is an ongoing challenge of modern cryptography. This work addresses the challenge, and constructs zero-knowledge arguments with sublinear communication complexity, and achievable computational demands. The security of our protocols is only based on the discrete logarithm assumption. Polynomial evaluation arguments are proposed for univariate polynomials, for multivariate polynomials, and for a batch of univariate polynomials. Furthermore, the polynomial evaluation argument is applied to construct practical membership and non-membership arguments. Finally, an efficient method for proving the correctness of a shuffle is proposed. The proposed protocols have been tested against current state of the art versions in order to verify their practicality in terms of run-time and communication cost. We observe that the performance of our protocols is fast enough to be practical for medium range parameters. Furthermore, all our verifiers have a better asymptotic behavior than earlier verifiers independent of the parameter range, and in real life settings our provers perform better than provers of existing protocols. The analysis of the results shows that the communication cost of our protocols is very small; therefore, our new protocols compare very favorably to the current state of the art

A Note On Groth-Ostrovsky-Sahai Non-Interactive Zero-Knowledge Proof System

In 2006, Groth, Ostrovsky and Sahai designed one non-interactive zero-knowledge (NIZK) proof system [new version, J. ACM, 59(3), 1-35, 2012] for plaintext being zero or one using bilinear groups with composite order. Based on the system, they presented the first perfect NIZK argument system for any NP language and the first universal composability secure NIZK argument for any NP language in the presence of a dynamic/adaptive adversary. This resolves a central open problem concerning NIZK protocols. In this note, we remark that in their proof system the prover has not to invoke the trapdoor key to generate witnesses. The mechanism was dramatically different from the previous works, such as Blum-Feldman-Micali proof system and Blum-Santis-Micali-Persiano proof system. We would like to stress that the prover can cheat the verifier to accept a false claim if the trapdoor key is available to him