On the Security of the Threshold Scheme Based on the Chinese Remainder Theorem

Abstract

Threshold schemes enable a group of users to share a secret by providing each user with a share. The scheme has a threshold t + 1 if any subset with cardinality t + 1 of the shares enables the secret to be recovered. In 1983, C. Asmuth and J. Bloom proposed such a scheme based on the Chinese remainder theorem. They derived a complex relation between the parameters of the scheme in order to satisfy some notion of security. However, at that time, the concept of security in cryptography had not yet been formalized. In this paper, we revisit the security of this threshold scheme in the modern context of security. In particular, we prove that the scheme is asymptotically optimal both from an information theoretic and complexity theoretic viewpoint when the parameters satisfy a simplified relationship. We mainly present three theorems, the two first theorems strengthen the result of Asmuth and Bloom and place it in a precise context, while the latest theorem is an improvement of a result obtained by Goldreich et al.

Lattice-Based Threshold Changeability for Standard Shamir Secret-Sharing Schemes

“…We will use the following definition of a threshold secret-sharing scheme, which is a slight modification of the definition in [17].…”

“…The correctness and security properties of a (t, n)-threshold secret-sharing scheme can be quantified by the following definitions, which are modifications of those in [17].…”

Sharing DSS by the Chinese Remainder Theorem

“…Let S be the adversarial coalition of size t − 1, and let y be the unique solution for y in Z M S . According to (1), Quisquater et al [14] showed that when moduli m i are chosen as consecutive primes 1 ≤ i ≤ n, the scheme has better security properties. In this paper, we assume that all m i are selected as consecutive primes.…”

Threshold cryptography based on Asmuth–Bloom secret sharing

“…However, this scheme is not exactly perfect since when t À 1 shares are known, the key candidates are not equally likely as described in Section 4. We refer the reader to a recent work by Quisquater et al [22] for a detailed security analysis of Asmuth-Bloom and some other Chinese Remainder Based SSSs.…”