Univariate polynomial factorization over finite fields with large extension degree

Abstract

The best known asymptotic bit complexity bound for factoring univariate polynomials over finite fields grows with $$d^{1.5 + o (1)}$$ for input polynomials of degree d, and with the square of the bit size of the ground field. It relies on a variant of the Cantor–Zassenhaus algorithm which exploits fast modular composition. Using techniques by Kaltofen and Shoup, we prove a refinement of this bound when the finite field has a large extension degree over its prime field. We also present fast practical algorithms for the case when the extension degree is smooth.