Modular Arithmetic Implementation with the Residue Number System ( RNS )

where SX is the size of X in bits and Xj is the j-th word of the binary representation of X split in k-bit words. Considering (1), in order to obtain the RNS representation of X, the constants 2 mod me,i can be precomputed and stored in the accelerator and the conversion is accomplished with a few RNS channel modular multiply-and-accumulate operations. For the reverse conversion, which means converting the computation results back to the binary representation, several approaches exist [3, 5, 7]. In the proposed framework, the conversion presented in [5] is adopted since it is suggested to be less complex and require lower dynamic ranges (smaller moduli sets) [1,4]. This conversion is based on the Chinese Remainder Theorem (CRT) that states