A quantum algorithm for computing multiplicative order of integers modulo a prime

Abstract

One of the challenging problems in number theory is to compute efficiently multiplicative order of integers modulo a prime. This problem was listed under open problems in the number theoretic complexity II proposed by Leonard M. Adleman and Kevin S.McCurley in 1994. This paper offers a polynomial-time quantum algorithm for computing the order k = min {x|x∈, ax ≡ 1(modp)}, if p∈ Primes and gcd(a,p) = 1, where a, p∈, and it requires 2 (1) 2 ((log p) +O )  quantum operations.