Applications of the montgomery exponent

We de£ne here the Montgomery Exponent of order s, modulo the odd integer N, by MEXP = MEXP(A, X, N, s) = A^X2^-s(X-1) (mod N), and illustrate some properties and usage of this operator. We show how A^X (mod N) can be obtained from MEXP(A, X, N, s) by one Montgomery multiplication. This suggests a new modular exponentiation algorithm that uses one Montgomery multiplication less than the number required with the standard method. This improves the performance, although the improvement is signi£cant only when the exponent X is short (e.g., modular squaring or RSA veri£cation). However, and even more important, this achieves code size reduction, which is appreciated when the exponentiation algorithm is written in a low level language and stored in (expensive) ROM. We also illustrate the potential advantage in performance and code size when known cryptographic applications are modi£ed in a way that MEXP replaces the standard modular exponentiation.