Paper 2026/2322

Understanding Sigma Protocols via MPC-in-the-Head

Abstract

In this paper, we propose an understanding of classical Sigma protocols (as denoted by $\Sigma$-protocols in the literature) through the lens of the MPC-in-the-head (MPCitH) paradigm. We seek not to construct new schemes, but to build a formal bridge connecting these two foundational areas. While many elegant yet foundational Sigma-protocols were introduced decades ago as standalone algebraic or combinatorial interactive proofs, reconsidering them via the modern aspect of (extended) MPCitH significantly enhances our cryptographic insight into their underlying structures. We present a formal mapping that translates classical commitment-challenge-and-response into the internal views of virtual parties, communicating via private point-to-point or public broadcast channels, secret-sharing schemes, commitments, and MPC consistency/correctness checks. By deconstructing algebraic schemes as linear secret-sharing protocols over public channels and combinatorial schemes as discrete threshold MPC instances over private/public channels, we isolate the exact structural reasons why their soundness and efficiency characteristics diverge. The protocols include: - Algebraic schemes: Schnorr, Okamoto, Chaum-Pedersen, Guillou-Quisquater, GMR for QR and discrete log over an unknown-order group. - Combinatorial schemes: Stern, GMW for Graph 3-Coloring, and Graph Isomorphism, among others. Although reinterpreting these already elegant, classic proofs using a heavy-weight machinery like MPCitH might seem redundant, it deepens our understanding of their underlying design principles.