As explained in my Reference Frame column of April 2007 (page 8), a key to factoring the product N = pq of two prime numbers, each hundreds of digits long, is being able to find the smallest power r for which ar (mod N) = 1 for random integers a. (Two numbers are equal modulo N if they differ by a multiple of N.) Peter Shor's 1994 discovery that a quantum computer would be superefficient at this cryptographically crucial task underlies today's wide-spread interest in quantum computation. Shor's period-finding algorithm—so called because the function f(x) = ax (mod N) is periodic with period r—illustrates in striking ways the novel basis quantum mechanics provides for computation.
In a quantum computer, a nonnegative integer x less than 2 n is represented by the product state
