Improved RNS Montgomery Modular Multiplication with Residue Recovery

Abstract

Finite field arithmetic in residue number system (RNS) necessitates modular reductions, which can be carried out with RNS Montgomery algorithm. By transforming long-precision modular multiplications into modular multiplications with small moduli, the computational complexity has decreased much. In this work, two implementation methods of RNS Montgomery algorithm, residue recovery as well as parallel base conversion, are reviewed and compared. Then, we propose a new residue recovery method that directly employs binary system rather than mixed radix system to perform RNS modular multiplications. This improvement is appropriate for a series of long-precision modular multiplications with variant operands, in which it is more efficient than parallel base conversion method.