MPC-in-the-Head Zero-Knowledge Proof for Rank Syndrome Decoding via Mixed-Field Secret Sharing

Quantum computing poses significant challenges to traditional zero-knowledge proof schemes based on number-theoretic assumptions. As a result, code-based cryptography has attracted increasing attention for its resistance against quantum computing. In this paper, we study the Rank Syndrome Decoding problem (RSD) and investigate its ZK proof formulation within the MPC-in-the-Head framework. To prove the possession of a secret witness, we reformulate the secret witness as a mixed-field matrix multiplication preserving the rank constraint, and then obtain a representation that aligns naturally with the local-view paradigm of MPC-in-the-Head. Utilizing this value-to-calculation technique, we introduce the RSD relation into a ZKBoo-style (2,3)-secret-sharing MPC-in-the-Head framework and obtain an RSD-based zero-knowledge proof scheme via mixed-field secret sharing. The resulting scheme reduces the proof size relative to generic formulations while preserving completeness, soundness, and zero-knowledge for the interactive protocol. The Fiat–Shamir non-interactive extension is analyzed only in the classical random oracle model; we do not claim QROM security for this variant.