A Multi-level Blocking Distinct Degree Factorization Algorithm

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)

Structured total least norm and approximate GCDs of inexact polynomials

The determination of an approximate greatest common divisor (GCD) of two inexact polynomials f=f(y) and g=g(y) arises in several applications, including signal processing and control. This approximate GCD can be obtained by computing a structured low rank approximation S*(f,g) of the Sylvester resultant matrix S(f,g). In this paper, the method of structured total least norm (STLN) is used to compute a low rank approximation of S(f,g), and it is shown that important issues that have a considerable effect on the approximate GCD have not been considered. For example, the established works only yield one matrix S*(f,g), and therefore one approximate GCD, but it is shown in this paper that a family of structured low rank approximations can be computed, each member of which yields a different approximate GCD. Examples that illustrate the importance of these and other issues are presented

Resolving zero-divisors using Hensel lifting

Algorithms which compute modulo triangular sets must respect the presence of zero-divisors. We present Hensel lifting as a tool for dealing with them. We give an application: a modular algorithm for computing GCDs of univariate polynomials with coefficients modulo a radical triangular set over the rationals. Our modular algorithm naturally generalizes previous work from algebraic number theory. We have implemented our algorithm using Maple's RECDEN package. We compare our implementation with the procedure RegularGcd in the RegularChains package.Comment: Shorter version to appear in Proceedings of SYNASC 201