Paper 2023/1420

Rogue-Instance Security for Batch Knowledge Proofs

Abstract

We propose a new notion of knowledge soundness, denoted rogue-instance security, for interactive and non-interactive batch knowledge proofs. Our notion, inspired by the standard notion of rogue-key security for multi-signature schemes, considers a setting in which a malicious prover is provided with an honestly-generated instance $x_1$, and may then be able to maliciously generate related "rogue" instances $x_2,\ldots,x_k$ for convincing a verifier in a batch knowledge proof of corresponding witnesses $w_1,\ldots,w_k$ for all $k$ instances - without actually having knowledge of the witness $w_1$ corresponding to the honestly-generated instance. This setting provides a powerful security guarantee for batch versions of a wide variety of practically-relevant protocols, such as Schnorr's protocol and similar ones. We present a highly-efficient generic construction of a batch proof-of-knowledge applicable to any algebraic Sigma protocols. The algebraic property refers to a homomorphic structure of the underlying group and includes Schnorr's protocol and others. We provide an almost tight security analysis for our generic batch protocol, which significantly improves upon the previously known security bounds even for the specific case of batch Schnorr protocol. We extend our results beyond algebraic Sigma protocols. We analyze the rogue-instance security of a general batch protocol with plus-one special soundness (a generalization of standard special soundness) and achieve improved security bounds in the generic case. Our results use a particular type of high-moment assumptions introduced by Rotem and Segev (CRYPTO 2021). These assumptions consider the hardness of a relation against algorithms with bounded expected running time. Although Rotem and Segev introduced these assumptions, they did not provide evidence to support their hardness. To substantiate and validate the high-moment assumptions, we present a new framework for assessing the concrete hardness of cryptographic problems against oracle algorithms with bounded expected runtime. Our framework covers generic models, including the generic group model, random oracle model, and more. Utilizing our framework, we achieve the first hardness result for these high-moment assumptions. In particular, we establish the second-moment hardness of the discrete-logarithm problem against expected-time algorithms in the generic group model.