Toward an optimal quantum algorithm for polynomial factorization over finite fields

Abstract

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 \cite{guo2016alg}.