Computing phi(N) for an RSA module with a single quantum query

Abstract

In this paper, we give a polynomial time algorithm to compute Phi(N) for an RSA module N using as input the order modulo N of a randomly chosen integer. This provides a new insight in the very important problem of factoring an RSA module with extra information. In fact, the algorithm is extremely simple and consists only on a computation of a greatest common divisor, two multiplications and a division. The algorithm works with a probability of at least 1-(1 )/( N 1/2-epsilon) , where E is any small positive constant.