Aryabhata Remainder Theorem for Moduli with Common Factors and Its Application in Information Protection Systems

The Chinese Remainder Theorem (CRT) can determine an integer from its residues modulo by a set of pairwise relatively prime moduli. For the requirement of flexibility, the CRT for moduli with common factors also has been proposed to deal with the case in which the moduli are not relatively prime. However, we discover that the previous schemes of CRT for moduli with common factors are incorrect while one modulus is the least common multiple of the other one. To solve this problem, we propose a new algorithm of Aryabhata Remainder Theorem (ART) for moduli with common factors in this paper. The proposed algorithm can be applied to any kind of moduli and its computation cost is less than that of the CRT-based algorithm. In addition, we also show how to apply the proposed method to the information protection systems in this paper.