Fast K\"otter-Nielsen-H{\o}holdt Interpolation over Skew Polynomial Rings and its Application in Coding Theory

TL;DR

This paper introduces a fast divide-and-conquer KNH interpolation algorithm for skew polynomials, enabling efficient decoding in various coding metrics without pre-processing of points.

Contribution

It presents a novel bottom-up KNH interpolation method that matches the speed of existing techniques while removing the need for pre-processing of interpolation points.

Findings

Achieves interpolation complexity of O(s^{8} M(n)) operations in F_{q^m}

Matches the speed of top-down minimal approximant bases techniques

Does not require pre-processing of interpolation points

Abstract

Skew polynomials are a class of non-commutative polynomials that have several applications in computer science, coding theory and cryptography. In particular, skew polynomials can be used to construct and decode evaluation codes in several metrics, like e.g. the Hamming, rank, sum-rank and skew metric. We propose a fast divide-and-conquer variant of K\"otter-Nielsen-H{\o}holdt (KNH) interpolation algorithm: it inputs a list of linear functionals on skew polynomial vectors, and outputs a reduced Gr\"obner basis of their kernel intersection. We show, that the proposed KNH interpolation can be used to solve the interpolation step of interpolation-based decoding of interleaved Gabidulin codes in the rank-metric, linearized Reed-Solomon codes in the sum-rank metric and skew Reed-Solomon codes in the skew metric requiring at most $\tilde{O}(s^{\omega} M(n))$ operations in $\mathbb{F}_{q^m}$ , where $n$ is the length of the code, $s$ the interleaving order, $M(n)$ the complexity for multiplying two skew polynomials of degree at most $n$, ${\omega}$ the matrix multiplication exponent and $\tilde{O}(\cdot)$ the soft-O notation which neglects log factors. This matches the previous best speeds for these tasks, which were obtained by top-down minimal approximant bases techniques, and complements the theory of efficient interpolation over free skew polynomial modules by the bottom-up KNH approach. In contrast to the top-down approach the bottom-up KNH algorithm has no requirements on the interpolation points and thus does not require any pre-processing.