Low-Rank Parity-Check Codes over the Ring of Integers Modulo a Prime Power

Renner, Julian; Puchinger, Sven; Wachter-Zeh, Antonia; Hollanti, Camilla; Freij-Hollanti, Ragnar

doi:10.1109/isit44484.2020.9174384

Cited by 11 publications

(26 citation statements)

References 8 publications

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“…Following the generalization of Gabidulin codes [4] over FPIR [6], it was recently shown in [7] that LRPC codes [5] can be defined over the small class of rings of integers modulo a prime power. The authors of [7] also provided a decoding algorithm together with an upper bound of its failure probability.…”

Section: Using the Smith Normal Form Of E(x)mentioning

confidence: 99%

“…Let us consider the ring Z m , where m ∈ N. If m is a prime power, then we know how to define LRPC over Z m [7]. We focus here on the case, where m is not a prime power.…”

Section: Generalization Of Lrpc Codes To the Ring Of Integers Modulo A Positive Integermentioning

confidence: 99%

“…The LRPC code is defined by a special parity-check matrix H. We suppose that the codeword c was transmitted and the word y = c + e is received, where e is the rank t error vector due to the channel. Since the error vector e is of rank t, its support is a free module of rank t. The error vector is taken among the free vectors ( f rk(e) = rk(e)) of S n m,s of rank t. The decoding algorithm, that generalizes the one in [7], is given in Algorithm 1.…”

Section: Lemma 37 Let M Be a Free R M -Submodule Of S Ms Of Dimension D Thenmentioning

confidence: 99%

“…The notion of rank metric has recently been extended to Finite Principal Ideal Rings (FPIR) by Kamche et al in [6], where they analyzed and proposed a decoder for Gabidulin codes over FPIR. Their work inspired the authors of [7] who defined LRPC codes over the small class of rings Z p r , where p is prime, which are particular FPIR. The authors of [7] conclude their work by introducing the problem of generalization of LRPC codes over a finite ring.…”

Section: Introductionmentioning

confidence: 99%

“…Their work inspired the authors of [7] who defined LRPC codes over the small class of rings Z p r , where p is prime, which are particular FPIR. The authors of [7] conclude their work by introducing the problem of generalization of LRPC codes over a finite ring.…”

Section: Introductionmentioning

confidence: 99%

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Generalization of low rank parity-check (LRPC) codes over the ring of integers modulo a positive integer

2021

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Following the work of Gaborit et al. (in: The international workshop on coding and cryptography (WCC 13), 2013) defining LRPC codes over finite fields, Renner et al. (in: IEEE international symposium on information theory, ISIT 2020, 2020) defined LRPC codes over the ring of integers modulo a prime power, inspired by the paper of Kamche and Mouaha (IEEE Trans Inf Theory 65(12):7718–7735, 2019) which explored rank metric codes over finite principal ideal rings. In this work, we successfully extend the work of Renner et al. by constructing LRPC codes over the ring $$\mathbb {Z}_{m}$$ Z m which is not a chain ring. We give a decoding algorithm and we study the failure probability of the decoder.

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Section: Using the Smith Normal Form Of E(x)mentioning

confidence: 99%

“…Let us consider the ring Z m , where m ∈ N. If m is a prime power, then we know how to define LRPC over Z m [7]. We focus here on the case, where m is not a prime power.…”

Section: Generalization Of Lrpc Codes To the Ring Of Integers Modulo A Positive Integermentioning

confidence: 99%

Section: Lemma 37 Let M Be a Free R M -Submodule Of S Ms Of Dimension D Thenmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Generalization of low rank parity-check (LRPC) codes over the ring of integers modulo a positive integer

2021

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Low-rank parity-check codes over finite commutative rings

Kamche,

Kalachi,

Djomou

et al. 2024

AAECC

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Low-rank parity-check codes over Galois rings

Renner

Neri

Puchinger

2020

Des. Codes Cryptogr.

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Low-rank parity-check (LRPC) codes are rank-metric codes over finite fields, which have been proposed by Gaborit et al. (Proceedings of the workshop on coding and cryptography WCC, vol 2013, 2013) for cryptographic applications. Inspired by a recent adaption of Gabidulin codes to certain finite rings by Kamche et al. (IEEE Trans Inf Theory 65(12):7718–7735, 2019), we define and study LRPC codes over Galois rings—a wide class of finite commutative rings. We give a decoding algorithm similar to Gaborit et al.’s decoder, based on simple linear-algebraic operations. We derive an upper bound on the failure probability of the decoder, which is significantly more involved than in the case of finite fields. The bound depends only on the rank of an error, i.e., is independent of its free rank. Further, we analyze the complexity of the decoder. We obtain that there is a class of LRPC codes over a Galois ring that can decode roughly the same number of errors as a Gabidulin code with the same code parameters, but faster than the currently best decoder for Gabidulin codes. However, the price that one needs to pay is a small failure probability, which we can bound from above.

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Low-Rank Parity-Check Codes over the Ring of Integers Modulo a Prime Power

Cited by 11 publications

References 8 publications

Generalization of low rank parity-check (LRPC) codes over the ring of integers modulo a positive integer

Generalization of low rank parity-check (LRPC) codes over the ring of integers modulo a positive integer

Low-rank parity-check codes over finite commutative rings

Low-rank parity-check codes over Galois rings

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