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Strengthening McEliece Cryptosystem [chapter]

Pierre Loidreau
2000 Lecture Notes in Computer Science  
McEliece cryptosystem is a public-key cryptosystem based on error-correcting codes. It constitutes one of the few alternatives to cryptosystems relying on number theory. We present a modification of the McEliece cryptosystem which strengthens its security without increasing the size of the public key. We show that it is possible to use some properties of the automorphism groups of the codes to build decodable patterns of large weight errors. This greatly strengthens the system against the decoding attacks.
doi:10.1007/3-540-44448-3_45 fatcat:h37h4cqf6vdldmnsyc5i2bjerq

Rank metric and Gabidulin codes in characteristic zero [article]

Gwezheneg Robert, Pierre Loidreau, Daniel Augot
2013 arXiv   pre-print
We transpose the theory of rank metric and Gabidulin codes to the case of fields of characteristic zero. The Frobenius automorphism is then replaced by any element of the Galois group. We derive some conditions on the automorphism to be able to easily transpose the results obtained by Gabidulin as well and a classical polynomial-time decoding algorithm. We also provide various definitions for the rank-metric.
arXiv:1305.4047v1 fatcat:by3ecimsf5cfdbquvsfljkz4gq

Generalized Gabidulin codes over fields of any characteristic [article]

Daniel Augot, Pierre Loidreau, Gwezheneg Robert
2017 arXiv   pre-print
Originally used for the decoding of Reed-Solomon codes, this algorithm was adapted to Gabidulin codes over finite fields by Loidreau [Loi06] .  ... 
arXiv:1703.09125v1 fatcat:tshniykevzae7go24gj62ggvqy

On circulant involutory MDS matrices

Victor Cauchois, Pierre Loidreau
2018 Designs, Codes and Cryptography  
We give a new algebraic proof of the non-existence of circulant involutory MDS matrices with coefficients in fields of characteristic 2. In odd characteristics we give parameters for the potential existence. If we relax circulancy to θ-circulancy, then there is no restriction to the existence of θ-circulant involutory MDS matrices even for fields of characteristic 2. Finally, we relax further the involutory definition and propose a new direct construction of almost involutory θ-circulant MDS
more » ... rices. We show that they can be interesting in hardware implementations. *
doi:10.1007/s10623-018-0520-3 fatcat:yxg2ofrgw5b5pgvvrbuydmaega

RAMESSES, a Rank Metric Encryption Scheme with Short Keys [article]

Julien Lavauzelle and Pierre Loidreau and Ba-Duc Pham
2019 arXiv   pre-print
We present a rank metric code-based encryption scheme with key and ciphertext sizes comparable to that of isogeny-based cryptography for an equivalent security level. The system also benefits from efficient encryption and decryption algorithms, which rely on linear algebra operations over finite fields of moderate sizes. The security only relies on rank metric decoding problems, and does not require to hide the structure of a code. Based on the current knowledge, those problems cannot be
more » ... ntly solved by a quantum computer. Finally, the proposed scheme admits a failure probability that can be precisely controlled and made as low as possible.
arXiv:1911.13119v1 fatcat:63cp4l7kyfcxtawkyxgwbawxi4

A New Rank Metric Codes Based Encryption Scheme [chapter]

Pierre Loidreau
2017 Lecture Notes in Computer Science  
We design a new McEliece-like rank metric based encryption scheme from Gabidulin codes. We explain why it is not affected by the invariant subspace attacks also known as Overbeck's attacks. The idea of the design mixes two existing approaches designing rank metric based encryption schemes. For a given security our public-keys are more compact than for the same security in the Hamming metric based settings.
doi:10.1007/978-3-319-59879-6_1 fatcat:kqdhhgfszbfinpdfbdu24af6wy

Skew codes of prescribed distance or rank

Lionel Chaussade, Pierre Loidreau, Felix Ulmer
2008 Designs, Codes and Cryptography  
In this paper we generalize the notion of cyclic code and construct codes as ideals in finite quotients of non commutative polynomial rings, so called skew polynomial rings of automorphism type. We propose a method to construct block codes of prescribed rank and a method to construct block codes of prescribed distance. Since there is no unique factorization in skew polynomial rings, there are much more ideals and therefore much more codes than in the commutative case. In particular we obtain a
more » ... 40, 23, 10] 4 code by imposing a distance and a [42, 14, 21] 8 code by imposing a rank, which both improve by one the minimum distance of the previously best known linear codes of equal length and dimension over those fields. There is a strong connection with linear difference operators and with linearized polynomials (or q-polynomials) reviewed in the first section.
doi:10.1007/s10623-008-9230-6 fatcat:h6jjmqikljgu5eiu2uwaphsfza

Rank metric and Gabidulin codes in characteristic zero

Daniel Augot, Pierre Loidreau, Gwezheneg Robert
2013 2013 IEEE International Symposium on Information Theory  
We transpose the theory of rank metric and Gabidulin codes to the case of fields of characteristic zero. The Frobenius automorphism is then replaced by any element of the Galois group. We derive some conditions on the automorphism to be able to easily transpose the results obtained by Gabidulin as well and a classical polynomial-time decoding algorithm. We also provide various definitions for the rank-metric.
doi:10.1109/isit.2013.6620278 dblp:conf/isit/AugotLR13 fatcat:z6jey3me2ngkxg42nalru7wwlu

Projected subcodes of the second order binary Reed-Muller code

Matthieu Legeay, Pierre Loidreau
2012 2012 IEEE International Symposium on Information Theory Proceedings  
In this paper we construct new subcodes of the second-order binary Reed-Muller code by using the permutation group and by projecting the code onto codes with smaller parameters. The permutation group of Reed-Muller codes is the general affine group and can be decomposed into the semi-direct product of the translation group and the general linear group. The action of the translation group projects the second order Reed-Muller code onto copies of the first order Reed-Muller code. The general
more » ... r group projects the code onto codes for which we can control the useful length and the dimension. These parameters depend on the dimension of the eigenspace of the chosen element of the general linear group for the eigenvalue 1.
doi:10.1109/isit.2012.6283977 dblp:conf/isit/LegeayL12 fatcat:u5hzjpxg3jftbcfbbngbuyvwzy

Using algebraic structures to improve LDPC code reconstruction over a noisy channel

Pierre Loidreau
2019 2019 IEEE International Symposium on Information Theory (ISIT)  
In this paper we show that algebraic structure of codes can be used to improve dramatically the efficiency of code reconstructions techniques especially in the case of quasicyclic LDPC codes of large block sizes which are widely used in standards. We focus on the case where the receiver gets noisy blocks, but the principle could be used in the case of non noisy reception. We also show that the smoother the length of the quasi-cycle is, the better the trade-off can be tuned.
doi:10.1109/isit.2019.8849756 dblp:conf/isit/Loidreau19 fatcat:s2mxi72jgbg7fmpf66ovnrdmkm

Direct construction of quasi-involutory recursive-like MDS matrices from 2-cyclic codes

Victor Cauchois, Pierre Loidreau, Nabil Merkiche
2017 IACR Transactions on Symmetric Cryptology  
A good linear diffusion layer is a prerequisite in the design of block ciphers. Usually it is obtained by combining matrices with optimal diffusion property over the Sbox alphabet. These matrices are constructed either directly using some algebraic properties or by enumerating a search space, testing the optimal diffusion property for every element. For implementation purposes, two types of structures are considered: Structures where all the rows derive from the first row and recursive
more » ... s built from powers of companion matrices. In this paper, we propose a direct construction for new recursive-like MDS matrices. We show they are quasi-involutory in the sense that the matrix-vector product with the matrix or with its inverse can be implemented by clocking a same LFSR-like architecture. As a direct construction, performances do not outperform the best constructions found with exhaustive search. However, as a new type of construction, it offers alternatives for MDS matrices design.
doi:10.46586/tosc.v2016.i2.80-98 fatcat:le5xtesp5ja4lgt75bbsqxj6qy

Direct construction of quasi-involutory recursive-like MDS matrices from 2-cyclic codes

Victor Cauchois, Pierre Loidreau, Nabil Merkiche
2017 IACR Transactions on Symmetric Cryptology  
A good linear diffusion layer is a prerequisite in the design of block ciphers. Usually it is obtained by combining matrices with optimal diffusion property over the Sbox alphabet. These matrices are constructed either directly using some algebraic properties or by enumerating a search space, testing the optimal diffusion property for every element. For implementation purposes, two types of structures are considered: Structures where all the rows derive from the first row and recursive
more » ... s built from powers of companion matrices. In this paper, we propose a direct construction for new recursive-like MDS matrices. We show they are quasi-involutory in the sense that the matrix-vector product with the matrix or with its inverse can be implemented by clocking a same LFSR-like architecture. As a direct construction, performances do not outperform the best constructions found with exhaustive search. However, as a new type of construction, it offers alternatives for MDS matrices design.
doi:10.13154/tosc.v2016.i2.80-98 dblp:journals/tosc/CauchoisLM16 fatcat:oq2f7wmah5hfjcbjib64d36ika

Randomized Decoding of Gabidulin Codes Beyond the Unique Decoding Radius [article]

Julian Renner, Thomas Jerkovits, Hannes Bartz, Sven Puchinger, Pierre Loidreau, Antonia Wachter-Zeh
2020 arXiv   pre-print
for instance the (modified) Faure-Loidreau system [9, 33] or the RAMESSES system [21] .  ...  For the modified Faure-Loidreau system, our algorithm provides the most efficient key recovery attack for one set of parameters, shown in Line 5 of Table 1 .  ... 
arXiv:1911.13193v3 fatcat:6pa6da7p75hmlca5zenhmk3qzu

An analysis of Coggia-Couvreur attack on Loidreau's rank-metric public key encryption scheme in the general case [article]

Pierre Loidreau, Ba-Duc Pham
2021
In 2017, Loidreau proposed a scheme based on Gabidulin codes masked with a small dimensional vector space [5] .  ... 
doi:10.48550/arxiv.2112.12445 fatcat:mnwhfsz4zvg4rkwolasttz3aw4

La pédopsychiatrie en héritage

Loïc Loidreau
2015 Empan  
Loidreau, op. cit. Philippe Mazet est professeur émérite de psychiatrie de l'enfant et de l'adolescent à la faculté de médecine Pitié-Salpêtrière (université Pierre et Marie Curie, Paris VI).  ...  Loidreau, op. cit. 9. M. Ruel, dans L. Loidreau, op. cit.  ... 
doi:10.3917/empa.100.0056 fatcat:yfks72qx5ve5pituoggji25l7q
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