Computing $\varphi(N)$ for an RSA module with a single quantum query

Abstract

In this paper we give a polynomial time algorithm to compute $\varphi(N)$ 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 $1-\frac{1}{N^{1/2-\epsilon}}$, where $\epsilon$ is any small positive constant.