Gil Segev

The Chou-Orlandi batch oblivious transfer (OT) protocol is a particularly attractive OT protocol that bridges the gap between practical efficiency and strong security guarantees and is especially notable due to its simplicity. The security analysis provided by Chou and Orlandi bases the security of their protocol on the hardness of the computational Diffie-Hellman (CDH) problem in prime-order groups. Concretely, in groups in which no better-than-generic algorithms are known for the CDH problem, their security analysis yields that an attacker running in time t and issuing q random-oracle queries breaks the security of their protocol with probability at most ϵ ≤ q 2 · t / 2 κ / 2 , where κ is the bit-length of the group's order. This concrete bound, however, is somewhat insufficient for 256-bit groups (e.g., for κ = 256 , it does not provide any guarantee already for t = 2 48 and q = 2 40 ). In this work, we establish a tighter concrete security bound for the Chou-Orlandi protocol. First, we introduce the list square Diffie-Hellman problem and present a tight reduction from the security of the protocol to the hardness of solving the list square Diffie-Hellman problem. That is, we completely shift the task of analyzing the concrete security of the protocol to that of analyzing the concrete hardness of the list square Diffie-Hellman problem. Second, we reduce the hardness of the list square Diffie-Hellman problem to that of the decisional Diffie-Hellman (DDH) problem without incurring a multiplicative loss. Our key observation is that although CDH and DDH have the same assumed concrete hardness, relying on the hardness of DDH enables our reduction to efficiently test the correctness of the solutions it produces. Concretely, in groups in which no better-than-generic algorithms are known for the DDH problem, our analysis yields that an attacker running in time t and issuing q ≤ t random-oracle queries breaks the security of the Chou-Orlandi protocol with probability at most ϵ ≤ t / 2 κ / 2 (i.e., we eliminate the above multiplicative q 2 term). We prove our results within the standard real-vs-ideal framework considering static corruptions by malicious adversaries, and provide a concrete security treatment by accounting for the statistical distance between a real-model execution and an ideal-model execution.

Bulletproofs, introduced by Bünz, Bootle, Boneh, Poelstra, Wuille and Maxwell (IEEE S& P, 2018), is a highly efficient non-interactive argument system that does not require a trusted setup. Recently, Bünz (PhD Thesis, 2023) extended Bulletproofs to support arguments for rank-1 constraint satisfaction (R1CS) systems, a widely-used representation for arithmetic satisfiability problems. Although the argument system constructed by Bünz preserves the attractive properties of Bulletproofs, it presents a gap between its completeness and soundness guarantees: The system is complete for a restricted set of instances, but sound only for a significantly broader set. Although argument systems for such gap relations nevertheless provide clear and concrete guarantees, the gaps they introduce may lead to various inconsistencies or undesirable gaps within proofs of security, especially when used as building blocks within larger systems. In this work we show that the argument system presented by Bünz can be extended to bridge the gap between its completeness and soundness, and to additionally provide honest-verifier zero-knowledge. For the extended argument system, we introduce a refined R1CS relation that captures the precise set of instances for which both completeness and soundness hold without resorting to a gap formulation. The extended argument system preserves the performance guarantees of the argument system presented by Bünz, and yields a non-interactive argument system using the Fiat-Shamir transform.

In this work we first present an explicit forking lemma that distills the information-theoretic essence of the high-moment technique introduced by Rotem and Segev (CRYPTO '21), who analyzed the security of identification protocols and Fiat-Shamir signature schemes. Whereas the technique of Rotem and Segev was particularly geared towards two specific cryptographic primitives, we present a stand-alone probabilistic lower bound, which does not involve any underlying primitive or idealized model. The key difference between our lemma and previous ones is that instead of focusing on the tradeoff between the worst-case or expected running time of the resulting forking algorithm and its success probability, we focus on the tradeoff between higher moments of its running time and its success probability. Equipped with our lemma, we then establish concrete security bounds for the BN and BLS multi-signature schemes that are significantly tighter than the concrete security bounds established by Bellare and Neven (CCS '06) and Boneh, Drijvers and Neven (ASIACRYPT '18), respectively. Our analysis does not limit adversaries to any idealized algebraic model, such as the algebraic group model in which all algorithms are assumed to provide an algebraic justification for each group element they produce. Our bounds are derived in the random-oracle model based on the standard-model second-moment hardness of the discrete logarithm problem (for the BN scheme) and the computational co-Diffie-Hellman problem (for the BLS scheme). Such second-moment assumptions, asking that the success probability of any algorithm in solving the underlying computational problems is dominated by the second moment of the algorithm's running time, are particularly plausible in any group where no better-than-generic algorithms are currently known.