Fast polynomial factorization over high algebraic extensions of finite fields

New algorithms are presented for factoring polynomials of degree n over the finite field of q elements, where q is a power of 2. When log q = n^1+a, where a>0 is constant, these algorithms are asymptotically faster than previous known algorithms, the fastest of which required time Ω(n(log q)²), or Ω(n^3+2a) in this case, which corresponds to the cost of computing x^q modulo an n degree polynomial. The new algorithms factor an arbitrary polynomial in time O(n^3+a+1+o(1)+n^2.69+1.69a). All measures are in fixed precision operations, that is in bit complexity. Moreover, in the special case where all the irreducible factors have the same degree, the new algorithms run in time O(n^2.69+1.69a). In particular, one may test a polynomial for irreducibility in O(n^2.69+1.69a) bit operations. These results generalize to the case where q = p^k, where p is a small, fixed prime.