A Linear Algebraic Threshold Essential Secret Image Sharing Scheme

Abstract

A secret sharing scheme allocates to each participant a share of a secret in such a way that authorized subsets of participants can reconstruct the secret, while shares of unauthorized subsets of participants provide no useful information about the secret. For positive integers r,s,t,n with r⩽s⩽t⩽n, an (r,s,t,n)–threshold essential secret sharing scheme is an algorithm that decomposes a secret S into n shares, s of which are essential, in a way that authorized subsets are precisely those with at least t members, at least r of whom are essential. This work proposes a lossless linear algebraic (r,s,t,n)–threshold essential secret image sharing scheme that decomposes the secret, S, into equally-sized shares, each of size 1/t the size of S. For each block, B, of S, the scheme assigns to the n participants distinct signature vectors v1,v2,…,vn in the vector space F2αt, where α is a suitable positive integer, typically between 2 and 5, inclusive. These signature vectors must adhere to some admissibility conditions in order to satisfy the secret sharing threshold properties. The decomposition of B into n shares is obtained by partitioning B into t vectors, then computing the share yj of the jth participant (1≤j≤n), as a linear combination of these parts with coefficients from the signature vj. The presented simulations showcase the effectiveness and robustness of the proposed scheme against standard statistical and security attacks. They further demonstrate its superiority with respect to existing schemes.