An adaptive exponentiation method

Exponentiation of large positive integer with a 512-bit exponent is the basis of several well-known cryptographic algorithms. In this paper, an adaptive method for improving the performance of the m-ary method is proposed and analyzed. Due to the efficient utilization of partial results, it is useful for systems with varied exponent and base. This method is based on two ideas. Firstly, for base jc, a few of the exponentiations with smaller exponents are precomputed on-line, that is, x²,x³, . . . ,x^(2w-1) are precomputed, where w is an optimization parameter. Secondly, a number of used partial results will be determined and stored in a look-up table during the computation. Assume that squaring is free as compared with multiplication, depending on the numbers of precomputations and partial results, the proposed method on average gives a 26-40% time reduction as compared with the m-ary method. On the other hand, it does require little temporary storage for the used partial results.