Algorithm 7: Shor [17]

Proceed with the following steps to retrieve a factor of a composite number with highprobability. Pick at random an integer $a\ne \pm 1$ modulo N. If GCD$(a,N)$ is a non-trivial factor of N, we have a solution. Otherwise generate, setup, and run the quantum Subroutine (also shown in Figure 42) to find a candidate for $s/r$. Use the continued fraction expansion to retrieve r (or a factor in r when s and r has a common factor).   If r is even, one of GCD$(a^{r/2}\pm 1,N)$ may be a non-trivial factor of N. This happens with highprobability.