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Cryptanalysis of Two McEliece Cryptosystems Based on Quasi-Cyclic Codes

Ayoub Otmani 1 Jean-Pierre Tillich 2 Léonard Dallot 1
1 Equipe AMACC - Laboratoire GREYC - UMR6072
GREYC - Groupe de Recherche en Informatique, Image, Automatique et Instrumentation de Caen
Abstract : We cryptanalyse here two variants of the McEliece cryptosystem based on quasi-cyclic codes. Both aim at reducing the key size by restricting the public and secret generator matrices to be in quasi-cyclic form. The first variant considers subcodes of a primitive BCH code. We prove that this variant is not secure by finding and solving a linear system satisfied by the entries of the secret permutation matrix.The other variant uses quasi-cyclic low density parity-check codes. This scheme was devised to be immune against general attacks working for McEliece type cryptosystems based on low density parity-check codes by choosing in the McEliece scheme more general one-to-one mappings than permutation matrices. We suggest here a structural attack exploiting the quasi-cyclic structure of the code and a certain weakness in the choice of the linear transformations that hide the generator matrix of the code. Our analysis shows that with high probability a parity-check matrix of a punctured version of the secret code can be recovered in cubic time complexity in its length. The complete reconstruction of the secret parity-check matrix of the quasi-cyclic low density parity-check codes requires the search of codewords of low weight which can be done with about 237 operations for the specific parameters proposed.
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https://hal.archives-ouvertes.fr/hal-01083566
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Submitted on : Monday, November 17, 2014 - 2:43:55 PM
Last modification on : Thursday, February 7, 2019 - 5:27:57 PM

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Ayoub Otmani, Jean-Pierre Tillich, Léonard Dallot. Cryptanalysis of Two McEliece Cryptosystems Based on Quasi-Cyclic Codes. Mathematics in Computer Science, Springer, 2010, 3 (2), pp.129-140. ⟨10.1007/s11786-009-0015-8⟩. ⟨hal-01083566⟩

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