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

We introduce a new family of rank metric codes: Low Rank Parity Check codes (LRPC), for which we propose an efficient probabilistic decoding algorithm. This family of codes can be seen as the equivalent of classical LDPC codes for the rank metric. We then use these codes to design cryptosystems à la...

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Veröffentlicht in:IEEE transactions on information theory Jg. 65; H. 12; S. 7697 - 7717
Hauptverfasser: Aragon, Nicolas, Gaborit, Philippe, Hauteville, Adrien, Ruatta, Olivier, Zemor, Gilles
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York IEEE 01.12.2019
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Institute of Electrical and Electronics Engineers
Schlagworte:
Algorithms
Codes
Computational Complexity
Computer Science
Computer systems
Cryptography
Cryptography and Security
Data encryption
Decoding
Encryption
Equivalence
Error correction codes
Information Theory
Iterative decoding
Mathematics
Measurement
Parity
Public key cryptography
Statistical analysis
ISSN:0018-9448, 1557-9654
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Zusammenfassung:We introduce a new family of rank metric codes: Low Rank Parity Check codes (LRPC), for which we propose an efficient probabilistic decoding algorithm. This family of codes can be seen as the equivalent of classical LDPC codes for the rank metric. We then use these codes to design cryptosystems à la McEliece: more precisely we propose two schemes for key encapsulation mechanism (KEM) and public key encryption (PKE). Unlike rank metric codes used in previous encryption algorithms -notably Gabidulin codes - LRPC codes have a very weak algebraic structure. Our cryptosystems can be seen as an equivalent of the NTRU cryptosystem (and also to the more recent MDPC code-based cryptosystem) in a rank metric context, due to the similar form of the public keys. The present paper is an extended version of the article introducing LRPC codes, with important new contributions. We have improved the decoder thanks to a new approach which allows for decoding of errors of higher rank weight, namely up to <inline-formula> <tex-math notation="LaTeX">\frac {2}{3}(n-k) </tex-math></inline-formula> when the previous decoding algorithm only decodes up to <inline-formula> <tex-math notation="LaTeX">\frac {n-k}{2} </tex-math></inline-formula> errors. Our codes therefore outperform the classical Gabidulin code decoder which deals with weights up to <inline-formula> <tex-math notation="LaTeX">\frac {n-k}{2} </tex-math></inline-formula>. This comes at the expense of probabilistic decoding, but the decoding error probability can be made arbitrarily small. The new approach can also be used to decrease the decoding error probability of previous schemes, which is especially useful for cryptography. Finally, we introduce ideal rank codes, which generalize double-circulant rank codes and allow us to avoid known structural attacks based on folding. To conclude, we propose different parameter sizes for our schemes and we obtain a public key of 3337 bits for key exchange and 5893 bits for public key encryption, both for 128 bits of security.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2019.2933535