Algorithms for comparison in residue number systems

Abstract

Residue Number Systems (RNSs) has been widely used in digital signal processing (DSP) systems and cases of fast computing, parallelism and fault tolerant because of its carry-free property. However, the comparison operation in an RNS is quite difficult and the computation cost is high, which are a significant limitation to apply it for division, scaling and overflow detection. Reverse conversions are used to the comparison operation in an RNS over last decades, where comparisons are carried out in a binary (weighted) system rather than an RNS. Chinese Remainder Theorem (CRT) and Mixed-radix Conversion (MRC) are the two main methods to perform residue-to-binary (R2B) conversion. However, the calculation of mix-radix coefficients in MRC scheme is a strictly sequential manner, while modular and multiplication operations related to large numbers are required in CRT scheme although they can be implemented in fully parallel. Especially, in order to satisfy the requirements of wide word binary operations such as public-key cryptography and modern digital signal processing (DSP), neither CRT nor MRC is a suitable solution for comparison in an RNS when the processed numbers are large enough. In the paper, novel comparison algorithms are proposed, which achieve comparison operation in an RNS directly without R2B reverse conversions. Even the average time complexity of the algorithm carried out in a sequential manner is O(1). To the best of our knowledge, the proposed algorithms first achieve all calculations for comparison in an RNS completed within each modulus channel in general cases.