Shor's Algorithm and Factoring: Don't Throw Away the Odd Orders

Shor’s algorithm factors an integer N in two steps. The quantum step computes the order of a mod N where a is relatively prime to N . The classical step uses this order to factor N . Descriptions of the classical step require the order, s, to be even and that a 6≡ −1 mod N . If s is odd or a ≡ −1 mod N , then the quantum step is repeated. This paper describes how each prime divisor of the order s, not just 2, can be used to factor N . 1 Sketch of Shor’s Algorithm Shor’s[4] algorithm factors a composite integer N , which is not a non-trivial power, in two steps. The first step uses quantum computing to find the order of some integer a modulo N , where gcd(a,N) = 1. In other words, this step finds the smallest positive integer s such that as ≡ 1 mod N . The second step uses the order, s, and classical techniques to factor N . If s is odd or as/2 ≡ −1 mod N , then the quantum step is repeated. Otherwise, let b2 ≡ as/2 mod N , and notice that b2 has order 2 modulo N . In other words, b 2 2 ≡ 1 mod N and ( b2 − 1 ) ≡ (b2 − 1) (b2 + 1) ≡ 0 mod N. A non-trivial factorization of N is N = gcd((b2 − 1) , N) gcd((b2 + 1) , N) .