Paper 2024/1414

Code-Based Zero-Knowledge from VOLE-in-the-Head and Their Applications: Simpler, Faster, and Smaller

Abstract

Zero-Knowledge (ZK) protocols allow a prover to demonstrate the truth of a statement without disclosing additional information about the underlying witness. Code-based cryptography has a long history but did suffer from periods of slow development. Recently, a prominent line of research have been contributing to designing efficient code-based ZK from MPC-in-the-head (Ishai et al., STOC 2007) and VOLE-in-the head (VOLEitH) (Baum et al., Crypto 2023) paradigms, resulting in quite efficient standard signatures. However, none of them could be directly used to construct privacy-preserving cryptographic primitives. Therefore, Stern's protocols remain to be the major technical stepping stones for developing advanced code-based privacy-preserving systems. This work proposes new code-based ZK protocols from VOLEitH paradigm for various relations and designs several code-based privacy-preserving systems that considerably advance the state-of-the-art in code-based cryptography. Our first contribution is a new ZK protocol for proving the correctness of a regular (non-linear) encoding process, which is utilized in many advanced privacy-preserving systems. Our second contribution are new ZK protocols for concrete code-based relations. In particular, we provide a ZK of accumulated values with optimal witness size for the accumulator (Nguyen et al., Asiacrypt 2019). Our protocols thus open the door for constructing more efficient privacy-preserving systems. Moreover, our ZK protocols have the advantage of being simpler, faster, and smaller compared to Stern-like protocols. To illustrate the effectiveness of our new ZK protocols, we develop ring signature (RS) scheme, group signature (GS) scheme, fully dynamic attribute-based signature scheme from our new ZK. The signature sizes of the resulting schemes are two to three orders of magnitude smaller than those based on Stern-like protocols in various parameter settings. Finally, our first ZK protocol yields a standard signature scheme, achieving ``signature size + public key size'' as small as $3.05$ KB, which is slightly smaller than the state-of-the-art signature scheme (Cui et al., PKC 2024) based on the regular syndrome decoding problems.