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. The algorithm consists only on a computation of a greatest common divisor, two multiplications and a division. The algorithm works with a probability of at least $1-\frac{C}{N^{1/2}}$.