Compact sequences of co-primes and their applications to the security of CRT-based threshold schemes

Asymptotically ideal Chinese remainder theorem ‐based secret sharing schemes for multilevel and compartmented access structures

“…We will prove that this new variant of the Asmuth‐Bloom secret sharing scheme is asymptotically ideal if and only if it is based on 1‐compact sequences of co‐primes [10] (this is similar to the result established in [20] on the original Asmuth‐Bloom secret sharing scheme).…”

“…Now we are ready to introduce the security concepts for the Asmuth-Bloom secret sharing scheme with public shares. We follow a similar line to the one in [8,10,20] and introduce the concepts of asymptotic perfectness, asymptotic information rate, and asymptotic idealness.…”

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

“…This result was later improved in [4] by showing that the asymptotic idealness of this scheme is achieved for a subclass of compact sequences of co-primes [4]. Compact sequences of co-primes capture very well the idea of sequence of numbers of the "same magnitude", and they are much denser than sequences of consecutive primes [4]. Moreover, [4] studies the security of the Asmuth-Bloom threshold scheme [1] and also proposes some asymptotically perfect and ideal variants of it.…”

“…They also proved that the threshold scheme in [3] is asymptotically ideal (and, therefore, asymptotically perfect) provided that it uses sequences of consecutive primes and the secret is uniformly chosen from the secret space. This result was later improved in [4] by showing that the asymptotic idealness of this scheme is achieved for a subclass of compact sequences of co-primes [4]. Compact sequences of co-primes capture very well the idea of sequence of numbers of the "same magnitude", and they are much denser than sequences of consecutive primes [4].…”

“…One of the techniques to construct threshold schemes is based on the Chinese Remainder Theorem (CRT) [1][2][3][4][5]. The main idea is to use sequences of pair-wise coprime positive integers with special properties.…”