Fast Kötter-Nielsen-Høholdt Interpolation Over Skew Polynomial Rings

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 the Kötter–Nielsen–Høholdt (KNH) interpolation algorithm: it inputs a list of linear functionals on skew polynomial vectors, and outputs a reduced Gröbner basis of their kernel intersection. This 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.