Shor's Algorithm

1. Pick a random number a < N
2. Compute gcd(a, N). This may be done using the Euclidean algorithm.
3. If gcd(a, N) ≠ 1, then there is a nontrivial factor of N, so we are done.
4. Otherwise, use the period-finding subroutine (below) to find r, the period of the following function:

   ,

   i.e. the smallest integer r for which f(x + r) = f(x).
5. If r is odd, go back to step 1.
6. If a ^r/2 ≡ -1 (mod N), go back to step 1.
7. The factors of N are gcd(a^r/2 ± 1, N). We are done.