A perfect secret sharing scheme for general access structures

Abstract

The secret sharing (SS) scheme has been widely used as the de facto paradigm for group key management in cryptography and distributed computation. An SS scheme for a general access structure (GSS) has drawn more and more attention since it allows flexible access control. However, previous solutions to GSS need to either assign multiple shares to each participant or to solve a complex nonlinear integer programming. In this paper, we propose a novel strategy to address the two problems simultaneously based on the Chinese Remainder Theorem (CRT) for a polynomial ring over a finite field. We classify general access structures satisfying the monotone property into the two families of maximum forbidden subsets and minimum qualified subsets conforming to the security condition and the revealing condition. The moduli used in the scheme are pairwise coprime polynomials and can take the irreducible polynomials. To find the degrees of these polynomials, we only need to solve an integer linear programming (ILP) by minimizing the sum of the degrees of all the moduli. The proposed scheme is inherently a weighted SS scheme and may have no solution which commonly exists in all GSS schemes. To ensure a solution, we put forth a preprocessing algorithm which separates the original access structure into several subaccess structures based on graph theory. The proposed scheme only assigns a share for each participant and achieves perfect security. It has good generalization and provides a universal approach to program the scheme construction for SS schemes with general access structures, threshold access structures and weighted access structures.