A Multi-level Blocking Distinct Degree Factorization Algorithm

TL;DR

This paper introduces a multi-level blocking algorithm for polynomial factorization over GF(2), significantly speeding up the search for irreducible trinomials and enabling the discovery of new primitive trinomials at very high degrees.

Contribution

The paper presents a novel multi-level blocking strategy that improves polynomial factorization efficiency, especially for sparse polynomials, and applies it to find high-degree primitive trinomials.

Findings

Achieved over 560x speedup over naive algorithms for degree 24,036,583 trinomials.

Discovered two new primitive trinomials of degree 24,036,583 over GF(2).

Developed a certificate for irreducibility that is faster to verify than full search.

Abstract

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).