Fast K\"otter-Nielsen-H\o holdt Interpolation over Skew Polynomial Rings

TL;DR

This paper introduces a fast divide-and-conquer algorithm for K"otter-Nielsen-H ext{o}holdt interpolation over skew polynomial rings, enhancing decoding efficiency for various advanced error-correcting codes used in cryptography.

Contribution

It presents a novel, efficient divide-and-conquer method for KNH interpolation over skew polynomial rings, applicable to decoding multiple types of codes in cryptography.

Findings

Significantly reduces interpolation complexity in decoding algorithms.

Applicable to Gabidulin, linearized Reed-Solomon, and skew Reed-Solomon codes.

Enhances decoding speed for cryptographic applications.

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. In this paper we propose a fast divide-and-conquer variant of the K\"otter-Nielsen-H{\o}holdt (KNH) interpolation over free modules over skew polynomial rings. The proposed KNH interpolation can be used to solve the interpolation step of interpolation-based decoding of (interleaved) Gabidulin, linearized Reed-Solomon and skew Reed-Solomon codes efficiently, which have various applications in coding theory and code-based quantum-resistant cryptography.