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.
