Subquadratic-time factoring of polynomials over finite fields

New probabilistic algorithms are presented for factoring univariate polynomials over finite fields. The algorithms factor a polynomial of degree n over a finite field of constant cardinality in time O(n 1.815 ). Previous algorithms required time Θ(n 2+o(1) ). The new algorithms rely on fast matrix multiplication techniques. More generally, to factor a polynomial of degree n over the finite field F q with q elements, the algorithms use O(n 1.815 log q) arithmetic operations in F q . The new baby step/giant step techniques used in our algorithms also yield new fast practical algorithms at super-quadratic asymptotic running time, and subquadratic-time methods for manipulating normal bases of finite fields.

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