Investigations on Fast Exponentiation Algorithms for RSA Cryptographic Applications

Abstract

Exponentiation is to compute XE for a positive integer E and modular exponentiation is to compute XE mod M for positive integers E and M. When the lengths of the operators are at least 1024 binary representations or 300 decimal digits, modular exponentiation can be time-consuming and is often the dominant part of the computation in many algebra systems. Since exponentiation is a sequence of multiplications, there are two kinds of methods to accelerate the speed of modular exponentiation. One is to reduce the number of multiplications and the other is to accelerate the multiplication itself. In this paper, we describe some efficient exponentiation methods, which can effectively reduce the number of multiplications and other methods, which can accelerate multiplication itself respectively. Most importantly, we also detailed analyze the computational complexity for two kinds of these methods respectively.