Secret sharing and duality

TL;DR

This paper investigates the duality in secret sharing schemes, showing that while many schemes have equal complexity with their duals, an almost perfect scheme with fewer participants can have lower complexity than its dual.

Contribution

It provides an almost complete answer to whether the complexity of secret sharing schemes equals that of their duals, introducing a nearly perfect scheme with fewer participants and lower complexity.

Findings

Existence of an almost perfect scheme with 174 participants

Such a scheme has lower complexity than its dual

Addresses a long-standing open problem in secret sharing

Abstract

Secret sharing is an important building block in cryptography. All explicitly defined secret sharing schemes with known exact complexity bounds are multi-linear, thus are closely related to linear codes. The dual of such a linear scheme, in the sense of duality of linear codes, gives another scheme for the dual access structure. These schemes have the same complexity, namely the largest share size relative to the secret size is the same. It is a long-standing open problem whether this fact is true in general: the complexity of any access structure is the same as the complexity of its dual. We give an almost answer to this question. An almost perfect scheme allows negligible errors, both in the recovery and in the independence. There exists an almost perfect ideal scheme on 174 participants whose complexity is strictly smaller than that of its dual.