(2025-12) New Constructions of Multiplicative Secret Sharing Schemes
2025-12-24
This paper investigates the multiplicative properties of linear codes in secret sharing schemes. To address the limitation that certain access structures cannot be realized by ideal linear codes, we introduce the notion of shortest linear codes as an ideal benchmark for code length. Since explicitly determining such shortest codes is generally computationally difficult, we propose an explicit construction that, for any given access structure, produces a length-efficient linear code whose induced the access structure. On this basis, we further define multiplicative ideal linear codes and multiplicative length-efficient linear codes, and derive necessary and sufficient conditions for the existence of multiplicativity. The effectiveness of the proposed approach is demonstrated by concrete examples. Compared with the construction of Cramer et al., the multiplicative linear codes obtained in this work have smaller length.