Distinguishing and Recovering Generalized Linearized Reed–Solomon Codes

Hörmann, Felicitas; Bartz, Hannes; Horlemann-Trautmann, Anna-Lena

doi:10.1007/978-3-031-29689-5_1

Cited by 3 publications

(2 citation statements)

References 23 publications

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“…A new approach is to consider the sum-rank metric which covers both the Hamming and the rank metric as special cases. Even though the gain in terms of key size might not be as large as for the rank metric, it is reasonable to hope that rank-metric attacks cannot be adapted to the sum-rank-metric case [21] and the corresponding systems will remain secure.…”

Section: Introductionmentioning

confidence: 99%

Interpolation-based decoding of folded variants of linearized and skew Reed–Solomon codes

Hörmann

Bartz

2023

Des. Codes Cryptogr.

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The sum-rank metric is a hybrid between the Hamming metric and the rank metric and suitable for error correction in multishot network coding and distributed storage as well as for the design of quantum-resistant cryptosystems. In this work, we consider the construction and decoding of folded linearized Reed–Solomon (FLRS) codes, which are shown to be maximum sum-rank distance (MSRD) for appropriate parameter choices. We derive an efficient interpolation-based decoding algorithm for FLRS codes that can be used as a list decoder or as a probabilistic unique decoder. The proposed decoding scheme can correct sum-rank errors beyond the unique decoding radius with a computational complexity that is quadratic in the length of the unfolded code. We show how the error-correction capability can be optimized for high-rate codes by an alternative choice of interpolation points. We derive a heuristic upper bound on the decoding failure probability of the probabilistic unique decoder and verify its tightness by Monte Carlo simulations. Further, we study the construction and decoding of folded skew Reed-Solomon codes in the skew metric. Up to our knowledge, FLRS codes are the first MSRD codes with different block sizes that come along with an efficient decoding algorithm.

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Section: Introductionmentioning

confidence: 99%

Interpolation-based decoding of folded variants of linearized and skew Reed–Solomon codes

Hörmann

Bartz

2023

Des. Codes Cryptogr.

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“…For the rank metric the problem of decoding beyond the unique decoding radius was addressed in [17] for Gabidulin codes. Known structural attacks for McEliece-like cryptosystem in the Hamming and Rank metric [18]- [20] have been generalized to the sum-rank metric [21]. This raises the question if the sum-rank metric can be adapted to other cryptosystems that are based on the hardness of decoding beyond the unique decoding radius, such as [22]- [24].…”

Section: Introductionmentioning

confidence: 99%

Randomized Decoding of Linearized Reed–Solomon Codes Beyond the Unique Decoding Radius

Jerkovits,

Bartz,

Wachter-Zeh

2023

2023 IEEE International Symposium on Information Theory (ISIT)

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In this paper we address the problem of decoding linearized Reed-Solomon (LRS) codes beyond their unique decoding radius. We analyze the complexity in order to evaluate if the considered problem is of cryptographic relevance, i.e., can be used to design cryptosystems that are computationally hard to break. We show that our proposed algorithm improves over other generic algorithms that do not take into account the underlying code structure.

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Fast Gao-Like Decoding of Horizontally Interleaved Linearized Reed–Solomon Codes

Hörmann,

Bartz

2023

Lecture Notes in Computer Science

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Both horizontal interleaving as well as the sum-rank metric are currently attractive topics in the field of code-based cryptography, as they could mitigate the problem of large key sizes. In contrast to vertical interleaving, where codewords are stacked vertically, each codeword of a horizontally s-interleaved code is the horizontal concatenation of s codewords of s component codes. In the case of horizontally interleaved linearized Reed-Solomon (HILRS) codes, these component codes are chosen to be linearized Reed-Solomon (LRS) codes. We provide a Gao-like decoder for HILRS codes that is inspired by the respective works for non-interleaved Reed-Solomon and Gabidulin codes. By applying techniques from the theory of minimal approximant bases, we achieve a complexity of Õ(s 2.373 n 1.635 ) operations in Fqm , where Õ(•) neglects logarithmic factors, s is the interleaving order and n denotes the length of the component codes. For reasonably small interleaving order s ≪ n, this is subquadratic in the component-code length n and improves over the only known syndrome-based decoder for HILRS codes with quadratic complexity. Moreover, it closes the performance gap to vertically interleaved LRS codes for which a decoder of complexity Õ(s 2.373 n 1.635 ) is already known. We can decode beyond the unique-decoding radius and handle errors of sum-rank weight up to s s+1 (n − k) for component-code dimension k. We also give an upper bound on the failure probability in the zero-derivation setting and validate its tightness via Monte Carlo simulations.

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Distinguishing and Recovering Generalized Linearized Reed–Solomon Codes

Cited by 3 publications

References 23 publications

Interpolation-based decoding of folded variants of linearized and skew Reed–Solomon codes

Interpolation-based decoding of folded variants of linearized and skew Reed–Solomon codes

Randomized Decoding of Linearized Reed–Solomon Codes Beyond the Unique Decoding Radius

Fast Gao-Like Decoding of Horizontally Interleaved Linearized Reed–Solomon Codes

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