Fast Root Finding for Interpolation-Based Decoding of Interleaved Gabidulin Codes

Bartz, Hannes; Jerkovits, Thomas; Puchinger, Sven; Rosenkilde, Johan

doi:10.1109/itw44776.2019.8989290

Cited by 5 publications

(22 citation statements)

References 26 publications

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“…Rank-metric codes are sets of matrices whose distance is measured by the rank of their difference. These codes and their most famous subclass, Gabidulin codes, were independently Vector Root Finding (Problem 14) for a set of ≤ + 1 skew polynomial vectors of dimension + 1, degree at most n, with degree constraints k (1) , . .…”

Section: B the Studied Codes And Their Historymentioning

confidence: 99%

“…We chose to rely on the PM-Basis algorithm over F q m [x] in the proof of Theorem 7 since it stresses the similarities and differences of the skew and ordinary polynomial case for approximant bases of order 1 of matrices of degree 0. For a self-contained proof of Theorem 7, which directly adapts the key ideas of the PM-Basis correctness proof in [74], we refer to the conference version of this paper [1].…”

Section: Algorithm 2: Rightskewbasecasementioning

confidence: 99%

“…Proof: Correctness and complexity follow directly from the correctness and complexity of Algorithm 6 (Theorem 22) and Algorithm 7 (Theorem 25) and the results in [5] (see also the brief summary above). We only need to be careful about two points: the entries of the first column of U are F q -linearly independent by definition of the α i ; also, Q (1) , . .…”

Section: A Computational Problems and Theirmentioning

confidence: 99%

“…. , α nt ] ∈ F nt q m be a vector whose entries are linearly independent over F q , and k = [k (1) , . .…”

Section: A Computational Problems and Theirmentioning

confidence: 99%

“…Let LIC Gab [ , α; n t , k] be a lifted interleaved Gabidulin code and U ⊆ F m( +1) q of dimension dim(U) = n r be a received subspace, given in form of a basis U ∈ F nr×( +1) q m with U = U q . The interpolation step of the Bartz-Wachter-Zeh decoder asks for a solution Q (1) , . .…”

Section: A Computational Problems and Theirmentioning

confidence: 99%

See 4 more Smart Citations

Fast Decoding of Codes in the Rank, Subspace, and Sum-Rank Metric

Bartz

Jerkovits

Puchinger

et al. 2021

IEEE Trans. Inform. Theory

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We speed up existing decoding algorithms for three code classes in different metrics: interleaved Gabidulin codes in the rank metric, lifted interleaved Gabidulin codes in the subspace metric, and linearized Reed-Solomon codes in the sumrank metric. The speed-ups are achieved by new algorithms that reduce the cores of the underlying computational problems of the decoders to one common tool: computing left and right approximant bases of matrices over skew polynomial rings. To accomplish this, we describe a skew-analogue of the existing PM-Basis algorithm for matrices over ordinary polynomials. This captures the bulk of the work in multiplication of skew polynomials, and the complexity benefit comes from existing algorithms performing this faster than in classical quadratic complexity. The new algorithms for the various decoding-related computational problems are interesting in their own and have further applications, in particular parts of decoders of several other codes and foundational problems related to the remainderevaluation of skew polynomials.

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Section: B the Studied Codes And Their Historymentioning

confidence: 99%

Section: Algorithm 2: Rightskewbasecasementioning

confidence: 99%

Section: A Computational Problems and Theirmentioning

confidence: 99%

“…. , α nt ] ∈ F nt q m be a vector whose entries are linearly independent over F q , and k = [k (1) , . .…”

Section: A Computational Problems and Theirmentioning

confidence: 99%

Section: A Computational Problems and Theirmentioning

confidence: 99%

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Fast Decoding of Codes in the Rank, Subspace, and Sum-Rank Metric

Bartz

Jerkovits

Puchinger

et al. 2021

IEEE Trans. Inform. Theory

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Fast Kötter–Nielsen–Høholdt interpolation over skew polynomial rings and its application in coding theory

Bartz,

Jerkovits,

Rosenkilde

2023

Des. Codes Cryptogr.

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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 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. 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 $${\widetilde{O}}\left( s^\omega {\mathfrak {M}}(n)\right) $$ O ~ s ω M ( n ) operations in $${\mathbb {F}}_{q^m}$$ F q m , where n is the length of the code, $$s$$ s the interleaving order, $${\mathfrak {M}}(n)$$ M ( n ) the complexity for multiplying two skew polynomials of degree at most n, $$\omega $$ ω the matrix multiplication exponent and $${\widetilde{O}}\left( \cdot \right) $$ O ~ · 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.

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Fast Decoding of Codes in the Rank, Subspace, and Sum-Rank Metric

Bartz¹,

Jerkovits²,

Puchinger³

et al. 2020

Preprint

Self Cite

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We speed up existing decoding algorithms for three code classes in different metrics: interleaved Gabidulin codes in the rank metric, lifted interleaved Gabidulin codes in the subspace metric, and linearized Reed-Solomon codes in the sum-rank metric. The speed-ups are achieved by reducing the core of the underlying computational problems of the decoders to one common tool: computing left and right approximant bases of matrices over skew polynomial rings. To accomplish this, we describe a skew-analogue of the existing PM-Basis algorithm for matrices over usual polynomials. This captures the bulk of the work in multiplication of skew polynomials, and the complexity benefit comes from existing algorithms performing this faster than in classical quadratic complexity. The new faster algorithms for the various decoding-related computational problems are interesting in their own and have further applications, in particular parts of decoders of several other codes and foundational problems related to the remainder-evaluation of skew polynomials.

show abstract

Fast Root Finding for Interpolation-Based Decoding of Interleaved Gabidulin Codes

Cited by 5 publications

References 26 publications

Fast Decoding of Codes in the Rank, Subspace, and Sum-Rank Metric

Fast Decoding of Codes in the Rank, Subspace, and Sum-Rank Metric

Fast Kötter–Nielsen–Høholdt interpolation over skew polynomial rings and its application in coding theory

Fast Decoding of Codes in the Rank, Subspace, and Sum-Rank Metric

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