How to detect whether Shor's algorithm succeeds against large integers without a quantum computer

Shor's algorithm is a well-known probabilistic method for factoring large composite integers in polynomial-time on a quantum computer. The method computes the order r of a random element x in the group Z(N)* and uses that information for splitting N with an application of the greatest common divisor algorithm. However, being probabilistic, the success of Shor's algorithm relies on some special properties of N. If r is even and x(r/2) not equivalent to -1 mod N, then gcd(x(r/2) - 1, N) reveals a nontrivial factor of N and the method succeeds. But even assuming that r is even and being given the complete prime factorization of N it is not obvious whether x(r/2) not equivalent to -1 mod N and, therefore, it is not easy to assert whether Shor's algorithm would split N without running it and looking at its answer. We present a strategy for detecting whether the splitting occurs without any need for running the quantum order-finding algorithm, but we must be given the prime factorization of N. This has allowed us to produce the first direct evidence of the probability of success of Shor's method. The composites chosen were the product of two randomly -generated probable primes of similar sizes that pass the Miller-Rabin test. (C) 2021 The Authors. Published by Elsevier B.V.