The Matrix Subcode Equivalence problem and its application to signature with MPC-in-the-Head

2025-07-21

AI Summary

This work initiates the first systematic study of two NP-complete problems—Matrix Subcode Equivalence (MSE) and Matrix Code Permutation (MCP)—both previously unexplored in cryptography. Building on MSE, we propose the first post-quantum digital signature scheme leveraging matrix subcode structure. Our construction employs the MPC-in-the-Head paradigm to design efficient zero-knowledge proofs, further optimized via Threshold-Computation-in-the-Head and VOLE-in-the-Head techniques to reduce protocol overhead. Theoretically, this is the first work to establish a formal connection between subcode structure and matrix code-based cryptography. Practically, our scheme achieves a 275-byte public key and a 4,800-byte signature, reducing the total public-key-plus-signature size by nearly 5× compared to CROSS, and yielding smaller signatures than both SPHINCS+ and MEDS. These results significantly advance parameter efficiency and practical deployability of code-based signatures.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

Nowadays, equivalence problems are widely used in cryptography, most notably to establish cryptosystems such as digital signatures, with MEDS, LESS, PERK as the most recent ones. However, in the context of matrix codes, only the code equivalence problem has been studied, while the subcode equivalence is well-defined in the Hamming metric. In this work, we introduce two new problems: the Matrix Subcode Equivalence Problem and the Matrix Code Permuted Kernel Problem, to which we apply the MPCitH paradigm to build a signature scheme. These new problems, closely related to the Matrix Code Equivalence problem, ask to find an isometry given a code $C$ and a subcode $D$. Furthermore, we prove that the Matrix Subcode Equivalence problem reduces to the Hamming Subcode Equivalence problem, which is known to be NP-Complete, thus introducing the matrix code version of the Permuted Kernel Problem. We also adapt the combinatorial and algebraic algorithms for the Matrix Code Equivalence problem to the subcode case, and we analyze their complexities. We find with this analysis that the algorithms perform much worse than in the code equivalence case, which is the same as what happens in the Hamming metric. Finally, our analysis of the attacks allows us to take parameters much smaller than in the Matrix Code Equivalence case. Coupled with the effectiveness of extit{Threshold-Computation-in-the-Head} or extit{VOLE-in-the-Head}, we obtain a signature size of $approx$ 4 800 Bytes, with a public key of $approx$ 275 Bytes. We thus obtain a reasonable signature size, which brings diversity in the landscape of post-quantum signature schemes, by relying on a new hard problem. In particular, this new signature scheme performs better than SPHINCS+, with a smaller size of public key + signature. Our signature compares also well with other signature schemes: compared to MEDS, the signature is smaller, and we reduced the size of the sum of signature and public key by a factor close to 5. We also obtain a signature size that is almost half the size of the CROSS signature scheme.

Problem

• Introduces Matrix Subcode Equivalence Problem for cryptography
• Reduces Matrix Subcode Equivalence to NP-Complete Hamming problem
• Develops post-quantum signature scheme with smaller size

Innovation

• Introduces Matrix Subcode Equivalence Problem
• Applies MPCitH paradigm for signatures
• Reduces signature size significantly