On the asymptotic idealness of the Asmuth-Bloom threshold secret sharing scheme

The Chinese remainder theorem (CRT) is a fundamental theorem in number theory, widely used in cryptography to design secret sharing schemes. The CRT-based secret sharing schemes proposed so far make use of sequences of pairwise co-prime integers with special properties. The way these sequences are chosen plays a crucial role in the security achieved by the schemes that rely on them. Moreover, the CRT-based secret sharing schemes could achieve at most asymptotic idealness. In this paper we prove that the Asmuth-Bloom threshold secret sharing scheme is asymptotic ideal if and only if it is based on 1-compact sequences of co-primes. Apart from this, a comprehensive analysis of the known variants of the Asmuth-Bloom threshold secret sharing scheme is provided, clarifying the security properties achieved by each of them.