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An Upper-Bound on the Decoding Failure Probability of the LRPC Decoder

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Cryptography and Coding (IMACC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14421))

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Abstract

Low Rank Parity Check (LRPC) codes form a class of rank-metric error-correcting codes that was purposely introduced to design public-key encryption schemes. An LRPC code is defined from a parity check matrix whose entries belong to a relatively low dimensional vector subspace of a large finite field. This particular algebraic feature can then be exploited to correct with high probability rank errors when the parameters are appropriately chosen. In this paper, we present theoretical upper-bounds on the probability that the LRPC decoding algorithm fails.

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Notes

  1. 1.

    We can also get this result by using directly Theorem 2 from [2].

  2. 2.

    Such t-tuples \(\boldsymbol{\textrm{a}}\) are of the form \((0,\dots ,0,a_i,\lambda _{i+1}a_i,\dots ,\lambda _ta_i)\) where i can take any value in \(\left\{ 1,\dots ,t \right\} \), \(a_i\) is any non-zero element in \(\mathcal {A}\), and \(\lambda _{i+1},\dots ,\lambda _t\) have arbitrary values in \(\mathbb {F}_q\). The number of such tuples is therefore at least \(\left( q^{d} - 1\right) \sum _{u=0}^{t-1} q^u\) because the choice over \(a_i\) can be restricted to the linear space of dimension d contained in the set \(\mathcal {A}\).

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Acknowledgments

E. Burle is supported by RIN100 program funded by Région Normandie. A. Otmani is supported by the grant ANR-22-PETQ-0008 PQ-TLS funded by Agence Nationale de la Recherche within France 2030 program, and by FAVPQC (EIG CONCERT-Japan & CNRS).

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  1. Univ Rouen Normandie, INSA Rouen Normandie, Université Le Havre Normandie, Normandie Univ, LITIS UR 4108, F-76000, Rouen, France

    Étienne Burle & Ayoub Otmani

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  1. Étienne Burle
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  2. Ayoub Otmani
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Correspondence to Ayoub Otmani .

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  1. University of London, Egham, UK

    Elizabeth A. Quaglia

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Burle, É., Otmani, A. (2024). An Upper-Bound on the Decoding Failure Probability of the LRPC Decoder. In: Quaglia, E.A. (eds) Cryptography and Coding. IMACC 2023. Lecture Notes in Computer Science, vol 14421. Springer, Cham. https://doi.org/10.1007/978-3-031-47818-5_1

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  • DOI: https://doi.org/10.1007/978-3-031-47818-5_1

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