Efficient exponentiation using precomputation and vector addition chains

Abstract

An efficient algorithm for exponentiation with precomputation is proposed and analyzed. Due to the precomputations, it is mainly useful for (discrete log based) systems with a. fixed base. The algorithm is an instance of a more general method. This method is based on two ideas. Firstly, the idea from [3] of splitting the exponentiation into the product of a number of exponentiations with smaller exponents. Secondly, the use of the technique of vector addition chains to compute this product of powers. Depending on the amount of precomputations and memory, the proposed algorithm is about two to six times as fast as binary square and multiply. It is slightly slower than the method from [3], but requires far less memory.