Factorization of Integers (original) (raw)

SHOR’S ALGORITHM Another way of factorizing 15 appears to have been demonstrated by Scientists at IBM’s Almaden Research Center in 2001. They built a small quantum computer and used the following algorithm due to Peter Shor. Assume N is composite. Choose a < N , gcd(a,N) = 1. Find r, the period of the function f : x 7→ a mod N ; that is, find the smallest r > 0 for which a ≡ 1 (mod N). Assume r is even and a 6≡ ±1 (mod N); otherwise start again with a new a. Then, since (a − 1)(a + 1) = a − 1 ≡ 0 (mod N), gcd(a − 1, N) is a non-trivial factor of N . For example, N = 15, a = 2 gives r = 4, gcd(2 − 1, 15) = 3 and gcd(2 + 1, 15) = 5. The determination of r is particularly suited to quantum computing. For details, see [2, section 8.5.2], or look up ‘Shor’s algorithm’ in Wikipedia.