Toward an Optimal Quantum Algorithm for Polynomial Factorization over Finite Fields.

We present a randomized quantum algorithm for polynomial factorization over finite fields. For polynomials of degree n over a finite field F-q, the average-case complexity of our algorithm is an expected O(n(1+o(1)) log(2+o(1)) q) bit operations. Only for a negligible subset of polynomials of degree n our algorithm has a higher complexity of O(n(4/3+o(1)) log(2+o(1)) q) bit operations. This breaks the classical 3/2-exponent barrier for polynomial factorization over finite fields [9].