Fast Quantum Algorithms of Solving an Instance of Quadratic Congruence on a Quantum Computer

It is assumed that P is the product of two prime numbers q₁ and q₂. If there is an integer 0 <; M <; P such that M² = C (mod P), i.e., the congruence has a solution, then C is said to be a quadratic congruence (mod P). Quadratic congruence (mod P) is a NP-complete problem. If the value of C is equal to one, then four integer solutions for M² = 1 (mod P) are, respectively, b, P - b, 1 and P - 1, where 1 <; P - b <; (P / 2) and (P / 2) <; b <; P - 1. This is a special case of quadratic congruence (mod P). In this paper, it is shown that four integer solutions for M² = 1 (mod P) can be found by means of the proposed quantum algorithms with polynomial quantum gates, polynomial quantum bits and the successful probability that is the same as that of Shor´s order-finding algorithm.