Sub-quadratic decoding of Gabidulin codes

Puchinger, Sven; Wachter-Zeh, Antonia

doi:10.1109/isit.2016.7541760

Cited by 17 publications

(21 citation statements)

References 16 publications

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“…The authors of the present paper gave several algorithms for multiplication in [3], with best complexityÕ(dr ω−1 ) achieved for d > r 2 . The most recent results by Puchinger and Wachter-Zeh [12] give a bound ofÕ(d ω+1 2 r) operations in K for multiplication in L[X, σ], which improves on the previous results [3] when d ∈ Θ(r), which is the most relevant case for applications in coding theory (see [12], §4.2). In the context of differential operators (which share many similarities with skew polynomials), Benoit, Bostan and Van der Hoeven have obtained a complexity ofÕ(min{d, r} ω−2 dr) (see [1], Theorem 1) for multiplication in L[x] ∂ .…”

Section: Introductionsupporting

confidence: 51%

“…We also show that our method can be used to improve the best known complexities of various related problems, such as multi-point evaluation, minimal subspace polynomial, and interpolation which are studied in [12]. We also improve the complexities for greatest common divisors and least common multiples.…”

Section: Introductionmentioning

confidence: 91%

“…, bn−1 . A generic fast algorithm for solving this problem has been proposed by Puchinger and Wachter-Zeh in [12], Theorem 26; it has complexityÕ(n max{log 2 (3), ω+1 2 } r) operations in K. In the special case we are considering, we shall see that this can be improved toÕ(nr).…”

Section: Evaluation and Interpolation At An Incomplete Normal Basismentioning

confidence: 96%

See 2 more Smart Citations

Fast Multiplication for Skew Polynomials

Caruso

Borgne

2017

Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation

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We describe an algorithm for fast multiplication of skew polynomials. It is based on fast modular multiplication of such skew polynomials, for which we give an algorithm relying on evaluation and interpolation on normal bases. Our algorithms improve the best known complexity for these problems, and reach the optimal asymptotic complexity bound for large degree. We also give an adaptation of our algorithm for polynomials of small degree. Finally, we use our methods to improve on the best known complexities for various arithmetics problems.

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Section: Introductionsupporting

confidence: 51%

Section: Introductionmentioning

confidence: 91%

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Fast Multiplication for Skew Polynomials

Caruso

Borgne

2017

Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation

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“…where rank(Z F E ′ F ) = 2 (unknown) and [Z F ] E2 ∈ F 7×2 q (known) consists of the columns of Z F with indices E 2 . Thus, we can correctly decode due to (15) and…”

Section: Improved Decoding Using Gmdmentioning

confidence: 99%

Space-time codes based on rank-metric codes and their decoding

Puchinger

Stern

Bossert

et al. 2016

2016 International Symposium on Wireless Communication Systems (ISWCS)

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Abstract-In this paper, a new class of space-time block codes is proposed. The new construction is based on finite-field rank-metric codes in combination with a rank-metric-preserving mapping to the set of Eisenstein integers. It is shown that these codes achieve maximum diversity order and improve upon existing constructions. Moreover, a new decoding algorithm for these codes is presented, utilizing the algebraic structure of the underlying finite-field rank-metric codes and employing latticereduction-aided equalization. This decoder does not achieve the same performance as the classical maximum-likelihood decoding methods, but has polynomial complexity in the matrix dimension, making it usable for large field sizes and numbers of antennas.

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“…An ℓ-shot Gabidulin code, that is, a Gabidulin code tailored for ℓn ′ outgoing links at the source, yields the maximum message size of ℓn ′ − 2t − ρ packets, but would require packet lengths m ≥ n = ℓn ′ instead of m ≥ n ′ , which may be impractically large. More importantly, decoding an ℓ-shot Gabidulin code using [26] would require O(n 2 ) operations over a field of size q ℓn ′ 0 (even faster decoders [27], [28] would be quite expensive for such field sizes, see Table I).…”

Section: Introductionmentioning

confidence: 99%

Reliable and Secure Multishot Network Coding Using Linearized Reed-Solomon Codes

Martínez-Peñas

Kschischang

2019

IEEE Trans. Inform. Theory

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Multishot network coding is considered in a worstcase adversarial setting in which an omniscient adversary with unbounded computational resources may inject erroneous packets in up to t links, erase up to ρ packets, and wire-tap up to µ links, all throughout ℓ shots of a linearly-coded network. Assuming no knowledge of the underlying linear network code (in particular, the network topology and underlying linear code may be random and change with time), a coding scheme achieving zero-error communication and perfect secrecy is obtained based on linearized Reed-Solomon codes. The scheme achieves the maximum possible secret message size of ℓn ′ − 2t − ρ − µ packets for coherent communication, where n ′ is the number of outgoing links at the source, for any packet length m ≥ n ′ (largest possible range). By lifting this construction, coding schemes for non-coherent communication are obtained with information rates close to optimal for practical instances. The required field size is q m , where q > ℓ, thus q m ≈ ℓ n ′ , which is always smaller than that of a Gabidulin code tailored for ℓ shots, which would be at least 2 ℓn ′ . A Welch-Berlekamp sum-rank decoding algorithm for linearized Reed-Solomon codes is provided, having quadratic complexity in the total length n = ℓn ′ , and which can be adapted to handle not only errors, but also erasures, wiretap observations and non-coherent communication. Combined with the obtained field size, the given decoding complexity is of O(n ′4 ℓ 2 log(ℓ) 2 ) operations in F2, whereas the most efficient known decoding algorithm for a Gabidulin code has a complexity of O(n ′3.69 ℓ 3.69 log(ℓ) 2 ) operations in F2, assuming a multiplication in a finite field F costs about log(|F|) 2 operations in F2.

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Sub-quadratic decoding of Gabidulin codes

Cited by 17 publications

References 16 publications

Fast Multiplication for Skew Polynomials

Fast Multiplication for Skew Polynomials

Space-time codes based on rank-metric codes and their decoding

Reliable and Secure Multishot Network Coding Using Linearized Reed-Solomon Codes

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