Function and secret sharing extensions for Blakley and Asmuth-Bloom secret sharing schemes

Abstract

Threshold cryptography deals with situations where the authority to initiate or perform cryptographic operations is distributed amongst a group of individuals. Usually in these situations a secret sharing scheme is used to distribute shares of a highly sensitive secret, such as the private key of a bank, to the involved individuals so that only when a sufficient number of them can reconstruct the secret but smaller coalitions cannot. The secret sharing problem was introduced independently by Blakley and Shamir in 1979. They proposed two different solutions. Both secret sharing schemes (SSS) are examples of linear secret sharing. Many extensions and solutions based on these secret sharing schemes have appeared in the literature, most of them using Shamir SSS. In this thesis, we apply these ideas to Blakley secret sharing scheme. Many of the standard operations of single-user cryptography have counterparts in threshold cryptography. Function sharing deals with the problem of distribution of the computation of a function (such as decryption or signature) among several parties. The necessary values for the computation are distributed to the participants using a secret sharing scheme. Several function sharing schemes have been proposed in the literature with most of them using Shamir secret sharing as the underlying SSS. In this work, we investigate how function sharing can be achieved using linear secret sharing schemes in general and give solutions of threshold RSA signature, threshold Paillier decryption and threshold DSS signature operations. The threshold RSA scheme we propose is a generalization of Shoup’s Shamir-based scheme. It is similarly robust and provably secure under the static adversary model. In threshold cryptography the authorization of groups of people are decided simply according to their size. There are also general access structures in which any group can be designed as authorized. Multipartite access structures constitute an example of general access structures in which members of a subset are equivalent to each other and can be interchanged. Multipartite access structures can be used to represent any access structure since all access structures are multipartite. To investigate secret sharing schemes using these access structures, we used Mignotte and Asmuth-Bloom secret sharing schemes which are based on the Chinese remainder theorem (CRT). The question we tried to asnwer was whether one can find a Mignotte or Asmuth-Bloom sequence for an arbitrary access structure. For this purpose, we adapted an algorithm that appeared in the literature to generate these sequences. We also proposed a new SSS which solves the mentioned problem by generating more than one sequence.