MICODE: A Minimal Code Design for Secret Sharing Scheme

Belkacem Imine, Rahul Saha, Rahul Saha, Mauro Conti

2025

Abstract

Secret Sharing Schemes (SSS) in cryptography often utilize minimal linear codes for efficiency, with minimum distance playing a crucial role. Determining minimal codewords in general linear codes presents a challenge known as the linear code covering problem. To address this, we propose MInimal COde DEsign (MICODE), a novel method for generating minimal codes from binary Reed-Muller (RM) codes using the Ashikhmin-Barg lemma. Unlike existing approaches limited to small RM codes, MICODE extends to higher orders through a systematic puncturing strategy. By recursively removing one-weight columns from the generator matrix, we reduce the RM code’s maximum Hamming weight preserving its minimum distance. The RM generator matrix’s structure, derived from the Kronecker product of lower-triangular binary matrices, facilitates this construction. We conduct rigorous mathematical analysis of MICODE establishing parameters for a secure SSS. While these minimal codes are unsuitable for error correction due to their reduced code rate, they are proven highly effective for cryptographic applications, such as Massey SSS, where security depends on minimum distance. Our analysis also explores trade-offs between code rate and error performance offering new insights into their theoretical and practical implications.

Paper Citation

in Harvard Style

Imine B., Saha R. and Conti M. (2025). MICODE: A Minimal Code Design for Secret Sharing Scheme. In Proceedings of the 22nd International Conference on Security and Cryptography - Volume 1: SECRYPT; ISBN 978-989-758-760-3, SciTePress, pages 381-391. DOI: 10.5220/0013517300003979

in Bibtex Style

@conference{secrypt25,
author={Belkacem Imine and Rahul Saha and Mauro Conti},
title={MICODE: A Minimal Code Design for Secret Sharing Scheme},
booktitle={Proceedings of the 22nd International Conference on Security and Cryptography - Volume 1: SECRYPT},
year={2025},
pages={381-391},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0013517300003979},
isbn={978-989-758-760-3},
}

in EndNote Style

TY - CONF

JO - Proceedings of the 22nd International Conference on Security and Cryptography - Volume 1: SECRYPT
TI - MICODE: A Minimal Code Design for Secret Sharing Scheme
SN - 978-989-758-760-3
AU - Imine B.
AU - Saha R.
AU - Conti M.
PY - 2025
SP - 381
EP - 391
DO - 10.5220/0013517300003979
PB - SciTePress