Univariate polynomial factorization over finite fields with large extension degree

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