We give a new algorithm for performing the distinct-degree factorization of a
polynomial P(x) over GF(2), using a multi-level blocking strategy. The coarsest
level of blocking replaces GCD computations by multiplications, as suggested by
Pollard (1975), von zur Gathen and Shoup (1992), and others. The novelty of our
approach is that a finer level of blocking replaces multiplications by
squarings, which speeds up the computation in GF(2)[x]/P(x) of certain interval
polynomials when P(x) is sparse. As an application we give a fast algorithm to
search for all irreducible trinomials x^r + x^s + 1 of degree r over GF(2),
while producing a certificate that can be checked in less time than the full
search. Naive algorithms cost O(r^2) per trinomial, thus O(r^3) to search over
all trinomials of given degree r. Under a plausible assumption about the
distribution of factors of trinomials, the new algorithm has complexity O(r^2
(log r)^{3/2}(log log r)^{1/2}) for the search over all trinomials of degree r.
Our implementation achieves a speedup of greater than a factor of 560 over the
naive algorithm in the case r = 24036583 (a Mersenne exponent). Using our
program, we have found two new primitive trinomials of degree 24036583 over
GF(2) (the previous record degree was 6972593)
