New modular multiplication algorithms for fast modular exponentiation

A modular exponentiation is one of the most important operations in public-key cryptography. However, it takes much time because the modular exponentiation deals with very large operands as 512-bit integers. The modular exponentiation is composed of repetition of modular multiplications. Therefore, we can reduce the execution time of it by reducing the execution time of each modular multiplication. In this paper, we propose two fast modular multiplication algorithms. One is for modular multiplications between different integers, and the other is for modular squarings. These proposed algorithms require single-precision multiplications fewer than those of Montgomery modular multiplication algorithms by 1/2 and 1/3 times, respectively. Implementing on PC, proposed algorithms reduce execution times by 50% and 30% compared with Montgomery algorithms, respectively.