Low Rank Parity Check Codes: New Decoding Algorithms and Applications to Cryptography

Aragon, Nicolas; Gaborit, Philippe; Hauteville, Adrien; Ruatta, Olivier; Zémor, Gilles

doi:10.48550/arxiv.1904.00357

Cited by 3 publications

(15 citation statements)

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“…Proof. The proof follows by the same arguments as in [4, Section 4.5] (see also [11] for more details), where the uniquedecoding property implies that H ext is full-rank and shows that there is at most one solution B ∈ R t×n q that solves the linear system of equations HB ⊤ a ⊤ = s, where the entries of a ∈ R t q,m form a basis of E. In the original papers about LRPC codes over finite fields, [1], [4], some of the properties of Definition 2 are used without explicitly stating them.…”

Section: Lemma 4 (Unique Erasure Decoding) Given a Parity-check Matri...mentioning

confidence: 99%

“…We will use the maximal-row-span property to prove a bound on the failure probability of the decoder in Section V. It is a sufficient condition that our bound (in particular Theorem 11 in Section V) holds. Although not explicitly stated, [4,Proposition 4.3] must also assume a similar or slightly weaker condition in order to hold. It does not hold for arbitrary paritycheck matrices as in [4, Definition 4.1] (see Remark 12 in Section V).…”

Section: Lemma 4 (Unique Erasure Decoding) Given a Parity-check Matri...mentioning

confidence: 99%

“…In contrast to Theorem 11 the full-row-span property was not assumed in [4,Proposition 4.3], which is the analogous statement for finite fields. However, also the statement in [4,Proposition 4.3] is only correct if we assume additional structure on the parity-check matrix.…”

Section: Suppose Further That H Has the Maximal-row-span And Unity Pr...mentioning

confidence: 99%

“…Lemma 13 (Equivalent of [4,Lemma 3.4]). Let A, B ⊆ R q,m be free R q -modules of dimensions α and β, respectively.…”

Section: Failure Of Intersection Conditionmentioning

confidence: 99%

“…They can be seen as the rank-metric analogs of low-density parity-check codes in the Hamming metric. In [4], new decoders for LRPC codes were proposed. Compared to other known rank-metric codes, LRPC codes have a comparably low minimum distance, but their decoding is efficient and they have a weak algebraic structure.…”

Section: Introductionmentioning

confidence: 99%

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

Renner¹,

Puchinger²,

Wachter-Zeh³

et al. 2020

Preprint

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We define and analyze low-rank parity-check (LRPC) codes over extension rings of the finite chain ring Zpr , where p is a prime and r is a positive integer. LRPC codes have originally been proposed by Gaborit et al. (2013) over finite fields for cryptographic applications. The adaption to finite rings is inspired by a recent paper by Kamche et al. (2019), which constructed Gabidulin codes over finite principle ideal rings with applications to space-time codes and network coding. We give a decoding algorithm based on simple linear-algebraic operations. Further, we derive an upper bound on the failure probability of the decoder.The upper bound is valid for errors whose rank is equal to the free rank.

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Section: Lemma 4 (Unique Erasure Decoding) Given a Parity-check Matri...mentioning

confidence: 99%

Section: Lemma 4 (Unique Erasure Decoding) Given a Parity-check Matri...mentioning

confidence: 99%

Section: Suppose Further That H Has the Maximal-row-span And Unity Pr...mentioning

confidence: 99%

“…Lemma 13 (Equivalent of [4,Lemma 3.4]). Let A, B ⊆ R q,m be free R q -modules of dimensions α and β, respectively.…”

Section: Failure Of Intersection Conditionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Renner¹,

Puchinger²,

Wachter-Zeh³

et al. 2020

Preprint

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Interleaving Loidreau’s Rank-Metric Cryptosystem

Renner

Puchinger

Wachter-Zeh

2019

2019 XVI International Symposium "Problems of Redundancy in Information and Control Systems" (REDUNDANCY)

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We propose and analyze an interleaved variant of Loidreau's rank-metric cryptosystem based on rank multipliers. We analyze and adapt several attacks on the system, propose design rules, and study weak keys. Finding secure instances requires near-MRD rank-metric codes which are not investigated in the literature. Thus, we propose a random code construction that makes use of the fact that short random codes over large fields are MRD with high probability. We derive an upper bound on the decryption failure rate and give example parameters for potential key size reduction.

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Low-Rank Parity-Check Codes over Galois Rings

Renner,

Neri,

Puchinger

2020

Preprint

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Low-rank parity-check (LRPC) are rank-metric codes over finite fields, which have been proposed by Gaborit et al. (2013) for cryptographic applications. Inspired by a recent adaption of Gabidulin codes to certain finite rings by Kamche et al. (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: New Decoding Algorithms and Applications to Cryptography

Cited by 3 publications

References 0 publications

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

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

Interleaving Loidreau’s Rank-Metric Cryptosystem

Low-Rank Parity-Check Codes over Galois Rings

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