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 ${\text{F}}_q$, 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 $O(n^{4 / 3 + o(1)} \log^{2
• o(1)}q)$bitoperations\mathrm{.}Thisbreakstheclassical$3/2$-exponent barrier for polynomial factorization over finite fields \cite{guo2016alg}.