New fast modular multiplication method and its application to modular exponentiation-based cryptography

In order to apply exponentiation-based cryptography, such as RSA cryptography and El Gamal cryptography, to a wide range of practical problems, it is desired to devise faster ciphering and deciphering processes. This paper proposes a new algorithm for improving the speed of the exponentiation-based computation. The proposed method is based on the idea in the exponentiation computation that the remainder in square/multiplication with modulus n is constructed from the remainders with moduli different from n. More precisely, the method is based on the following two ideas. (1) The remainder in regard to n can be constructed from the remainder with modulus n + 1 and the remainder with modulus n + 2. (2) It often happens that n + 1 and n + 2 can easily be factorized, even if n is a prime number or difficult to be factorized into prime factors. Then, the Chinese remainder theorem can be applied to the remainder calculation with those numbers as the moduli. The bit computational complexity of the proposed method is estimated, and it is shown, assuming the parallel computation, that the computational complexity is less than in the conventional method. Especially when n + 1 and n + 2 are factorized almost uniformly into K factors, the computational complexity asymptotically follows 1/K. The proposed method will be useful not only in the cryptography requiring the exponentiation computation, but also in improving the speed of the signal processing that requires similar computations.