Enabling Efficient Applications of CRT and GCRT in Practice

As celebrated algorithms, Chinese Remainder Theorem (CRT) and Generalized Chinese Remainder Theorem (GCRT) are widely used in many fields. Generating pairwise coprime moduli efficiently is critical to constructing CRT system. Bernstein's recursive algorithm is linear-time in theory but on the condition that at least linear-number of processors are required to cooperate in parallel. Therefore, it is not practical. To enable efficient application of CRT and GCRT in practice, Algorithm 1 and 2 are first devised for generating pairwise coprime elements, both with time complexity. In functionality, Algorithm 1 generates any number of pairwise coprime elements continuously, thus it is highly scalable. In contrast, Algorithm 2 converts elements into pairwise coprime elements. Experiments on common laptops show both algorithm are much more efficient than Bernstein's algorithm. Next, Algorithm 2 is applied to construct a CRT system having the same solution as a given GCRT system with time complexity, which is a substantial and efficient complement to Szabo- Tanaka's GCRT algorithm. Finally, based on the idea of Algorithm 2, a parallel GCRT algorithm is proposed to support distributed computing, which allows each party to evaluate its component independently and thus is efficient in communication. Moreover, its time complexity suggests that it is more efficient in computation than the famous Ore's GCRT algorithm with time complexity.