Realizing secret sharing with general access structure

Abstract

Secret sharing (SS) is one of the most important cryptographic primitives used for data outsourcing. The (t, n) SS was introduced by Shamir and Blakley separately in 1979. The secret sharing policy of the (t, n) threshold SS is far too simple for many applications because it assumes that every shareholder has equal privilege to the secret or every shareholder is equally trusted. Ito et al. introduced the concept of a general secret sharing scheme (GSS). In a GSS, a secret is divided among a set of shareholders in such a way that any “qualified” subset of shareholders can access the secret, but any “unqualified” subset of shareholders cannot access the secret. The secret access structure of GSS is far more flexible than threshold SS. In this paper, we propose an optimized implementation of GSS. Our proposed scheme first uses Boolean logic to derive two important subsets, one is called Min which is the minimal positive access subset and the other is called Max which is the maximal negative access subset, of a given general secret sharing structure. Then, conditions of parameters of a GSS are established based on these two important subsets. Furthermore, integer linear/non-linear programming is used to optimize the size of shares of a GSS. The complexity of linear/non-linear programming is O(n), where n is the number of shares generated by the dealer. This proposed design can be applied to implement GSS based on any classical SS. However, our proposed method is limited to be applicable to some general secret sharing policies. We use two GSSs, one is based on Shamir's weighted SS (WSS) using linear polynomial and the other is based on Asmuth-Bloom's SS using Chinese Remainder Theorem (CRT), to demonstrate our design. In comparing with existing GSSs, our proposed scheme is more efficient and can be applied to all classical SSs.