A new faster algorithm for factoring skew polynomials over finite fields

Abstract

International audienceIn this paper, we provide an algorithm for the factorization of skew polynomials over finite fields. It is faster than the previously known algorithm, which was due to Giesbrecht (1998). There are two main improvements. The first one is obtained through a careful study of the structure of the quotients of a skew polynomial ring, using theoretical results relating skew polynomial rings and Azumaya algebras. The second improvement is provided by giving faster sub-algorithms for the arithmetic in skew polynomial rings, such as multiplication, division, and extended Euclidean division

Sub-quadratic decoding of Gabidulin codes

“…However, the division of two linearized polynomials was so far believed to be in O(s 2 ), compare [5]. We show that the reduction of linearized polynomial division to skew polynomial multiplication in [6] implies a sub-quadratic division algorithm by generalizing the above mentioned multiplication algorithm to skew polynomials. Finding a minimal subspace polynomial and performing a multi-point evaluation were both known to have complexity O(s 2 ), see [7], and the interpolation O(s 3 ).…”

Matroidal root structure of skew polynomials over finite fields

“…where deg r r (x) < deg g(x) or r r (x) = 0. An algorithm for this is also laid out in [4], while the problem of fast factoring in general is addressed in [9] and [5].…”

“…For the other direction, the proof is essentially the same as in the right case. The key note is that d i=1 f i x q m−1 −1 is an (m − 1)-linearized polynomial, and so by Theorem 4 of [15], there are also only q d solutions to (5), so once we find d F q -linearly independent b i 's, we must have that all roots are a (q m−1 − 1)th power of some F q -linear combination of them. Proof.…”

Fast Multiplication for Skew Polynomials

“…Classically, fast multiplication algorithms can be used to speed up many other computations. This general philosophy works for skew polynomials as well and was concretized in [3], §3.2. Below, we analyze briefly the impact of the algorithms designed above in this paper.…”

“…Euclidean division. An algorithm that performs (right) Euclidean divisions in L[X, σ] and takes advantage of fast multiplication algorithm is depicted in [3], §3.2.1 (Algorithm REuclideanDivision). Proposition 3.2.3 of loc.…”