22 research outputs found
Generalized Subspace Subcodes in the Rank Metric
Rank-metric codes were studied by E. Gabidulin in 1985 after a brief
introduction by Delsarte in 1978 as an equivalent of Reed-Solomon codes, but
based on linearized polynomials. They have found applications in many areas,
including linear network coding and space-time coding.
They are also used in cryptography to reduce the size of the keys compared to
Hamming metric codes at the same level of security. However, some families of
rank-metric codes suffer from structural attacks due to the strong algebraic
structure from which they are defined.
It therefore becomes interesting to find new code families in order to
address these questions in the landscape of rank-metric codes.
\par In this paper, we provide a generalization of Subspace Subcodes in Rank
metric introduced by Gabidulin and Loidreau. We also characterize this family
by giving an algorithm which allows to have its generator and parity-check
matrices based on the associated extended codes. We have also studied the
specific case of Gabidulin codes whose underlying decoding algorithms are
known. Bounds for the cardinalities of these codes, both in the general case
and in the case of Gabidulin codes, are also provided
Design, optimization and Real Time implementation of a new Embedded Chien Search Block for Reed-Solomon (RS) and Bose-Chaudhuri-Hocquenghem (BCH) codes on FPGA Board
The development of error correcting codes has been a major concern for communications systems. Therefore, RS and BCH (Reed-Solomon and Bose, Ray-Chaudhuri and Hocquenghem) are effective methods to improve the quality of digital transmission. In this paper a new algorithm of Chien Search block for embedded systems is proposed. This algorithm is based on a factorization of error locator polynomial. i.e, we can minimize an important number of logic gates and hardware resources using the FPGA card. Consequently, it reduces the power consumption with a percentage which can reach 40 % compared to the basic RS and BCH decoder. The proposed system is designed, simulated using the hardware description language (HDL) and Quartus development software. Also, the performance of the designed embedded Chien search block for decoder RS\BCH (255, 239) has been successfully verified by implementation on FPGA board
Expanded Gabidulin Codes and Their Application to Cryptography
This paper presents a new family of linear codes, namely the expanded
Gabidulin codes. Exploiting the existing fast decoder of Gabidulin codes, we
propose an efficient algorithm to decode these new codes when the noise vector
satisfies a certain condition. Furthermore, these new codes enjoy an excellent
error-correcting capability because of the optimality of their parent Gabidulin
codes. Based on different masking techniques, we give two encryption schemes by
using expanded Gabidulin codes in the McEliece setting. According to our
analysis, both of these two cryptosystems can resist the existing structural
attacks. Our proposals have an obvious advantage in public-key representation
without using the cyclic or quasi-cyclic structure compared to some other
code-based cryptosystems
Fast Decoding of Interleaved Linearized Reed-Solomon Codes and Variants
We construct s-interleaved linearized Reed-Solomon (ILRS) codes and variants
and propose efficient decoding schemes that can correct errors beyond the
unique decoding radius in the sum-rank, sum-subspace and skew metric. The
proposed interpolation-based scheme for ILRS codes can be used as a list
decoder or as a probabilistic unique decoder that corrects errors of sum-rank
up to , where s is the interleaving order, n the
length and k the dimension of the code. Upper bounds on the list size and the
decoding failure probability are given where the latter is based on a novel
Loidreau-Overbeck-like decoder for ILRS codes. The results are extended to
decoding of lifted interleaved linearized Reed-Solomon (LILRS) codes in the
sum-subspace metric and interleaved skew Reed-Solomon (ISRS) codes in the skew
metric. We generalize fast minimal approximant basis interpolation techniques
to obtain efficient decoding schemes for ILRS codes (and variants) with
subquadratic complexity in the code length. Up to our knowledge, the presented
decoding schemes are the first being able to correct errors beyond the unique
decoding region in the sum-rank, sum-subspace and skew metric. The results for
the proposed decoding schemes are validated via Monte Carlo simulations.Comment: submitted to IEEE Transactions on Information Theory, 57 pages, 10
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On McEliece type cryptosystems using self-dual codes with large minimum weight
One of the finalists in the NIST post-quantum cryptography competition is the Classic McEliece cryptosystem.
Unfortunately, its public key size represents a practical limitation. One option to address this problem is to use different families of error-correcting codes. Most of such attempts failed as those cryptosystems were proved not secure.
In this paper, we propose a McEliece type cryptosystem using high minimum distance self-dual codes and punctured codes derived from them. To the best of our knowledge, such codes have not been implemented in a code-based cryptosystem until now.
For the 80-bit security case, we construct an optimal self-dual code of length 1\,064, which, as far as we are aware, was not presented before. Compared to the original McEliece cryptosystem, this allows us to reduce the key size by about 38.5\%
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