The Calcualtion of Multiplicative Inverses Over GF(P) Efficiently Where P is a Mersenne Prime

The extended Euclidean algorithm is typically used to calculate multiplicative inverses over finite fields and rings of integers. The algorithm presented here has approximately the same number of average iterations and maximum number of iterations. It is shown, when P is a Mersenne prime, implementation of this algorithm on a processor, designed especially for mod P arithmetic operations, produces a more efficient algorithm with respect to the amount of program statements and number of operations. It is then shown heuristically, when the division and multiplications are performed simultaneously, the Euclidean algorithm has fewer subiterations.