Fast Operations on Linearized Polynomials and their Applications in Coding Theory

TL;DR

This paper introduces fast algorithms for operations on linearized polynomials, significantly improving computational efficiency and enabling sub-quadratic decoding of Gabidulin codes, with broad applications in coding theory.

Contribution

A novel multiplication algorithm for skew polynomials with sub-quadratic complexity, leading to faster algorithms for key operations and decoding in coding theory.

Findings

Faster multiplication algorithm for skew polynomials when s ≤ m

Sub-quadratic division algorithm for linearized polynomials

First sub-quadratic error and erasure decoding for Gabidulin codes

Abstract

This paper considers fast algorithms for operations on linearized polynomials. We propose a new multiplication algorithm for skew polynomials (a generalization of linearized polynomials) which has sub-quadratic complexity in the polynomial degree $s$, independent of the underlying field extension degree~$m$. We show that our multiplication algorithm is faster than all known ones when $s \leq m$. Using a result by Caruso and Le Borgne (2017), this immediately implies a sub-quadratic division algorithm for linearized polynomials for arbitrary polynomial degree $s$. Also, we propose algorithms with sub-quadratic complexity for the $q$-transform, multi-point evaluation, computing minimal subspace polynomials, and interpolation, whose implementations were at least quadratic before. Using the new fast algorithm for the $q$-transform, we show how matrix multiplication over a finite field can be implemented by multiplying linearized polynomials of degrees at most $s=m$ if an elliptic normal basis of extension degree $m$ exists, providing a lower bound on the cost of the latter problem. Finally, it is shown how the new fast operations on linearized polynomials lead to the first error and erasure decoding algorithm for Gabidulin codes with sub-quadratic complexity.