Paper 2025/2295 Original: eprint.iacr.org/2025/2295 ↗

An Ideal Linear Secret Sharing Scheme for Complete $t$-Partite $k$-Uniform Hypergraph Access Structures

Abstract

Secret sharing schemes represent a crucial cryptographic protocol, with linear codes serving as a primary tool for their construction. This paper systematically investigates the construction of ideal secret sharing schemes for complete $t$-partite $k$-uniform hypergraph access structures using linear codes as the tool. First, it is proved that the generator matrix $G$ of an ideal linear code realizing a complete $t$-partite $2$-uniform hypergraph access structure must have a rank of $2$. Simultaneously, a novel method for constructing an ideal secret sharing scheme that realizes such access structures is proposed. Building on this foundation, the case of complete $t$-partite $2$-uniform hypergraphs is extended to complete $t$-partite $k$-uniform hypergraphs, and a method for constructing ideal secret sharing schemes to realize them is provided. Compared with existing approaches, both Shamir’s method and the scheme proposed by Brickell et al. are special cases of our proposed approach.