Fast Parallel Montgomery Binary Exponentiation Algorithm Using Canonical- Signed-Digit Recoding Technique

In this paper, a new parallel Montgomery binary exponentiation algorithm was proposed. This algorithm is based on the Montgomery modular reduction technique, binary method, common-multiplicand-multiplication (CMM) algorithm, and the canonical-signed-digit recoding (CSD) technique. By using the CMM algorithm of computing the common part from two modular multiplications, the same common part in two modular multiplications can be computed once rather twice, we can thus improve the efficiency of the binary exponentiation algorithm by decreasing the number of modular multiplications. Furthermore, by using the proposed parallel CMM-CSD Montgomery binary exponentiation algorithm, the total number of single-precision multiplications can be reduced by about 66.7% and 30% as compared with the original Montgomery algorithm and the Ha-Moon’s improved Montgomery algorithm, respectively.