84 research outputs found
Structural Properties of Twisted Reed-Solomon Codes with Applications to Cryptography
We present a generalisation of Twisted Reed-Solomon codes containing a new
large class of MDS codes. We prove that the code class contains a large
subfamily that is closed under duality. Furthermore, we study the Schur squares
of the new codes and show that their dimension is often large. Using these
structural properties, we single out a subfamily of the new codes which could
be considered for code-based cryptography: These codes resist some existing
structural attacks for Reed-Solomon-like codes, i.e. methods for retrieving the
code parameters from an obfuscated generator matrix.Comment: 5 pages, accepted at: IEEE International Symposium on Information
Theory 201
Codes, Cryptography, and the McEliece Cryptosystem
Over the past several decades, technology has continued to develop at an incredible rate, and the importance of properly securing information has increased significantly. While a variety of encryption schemes currently exist for this purpose, a number of them rely on problems, such as integer factorization, that are not resistant to quantum algorithms. With the reality of quantum computers approaching, it is critical that a quantum-resistant method of protecting information is found. After developing the proper background, we evaluate the potential of the McEliece cryptosystem for use in the post-quantum era by examining families of algebraic geometry codes that allow for increased security. Finally, we develop a family of twisted Hermitian codes that meets the criteria set forth for security
Further Generalisations of Twisted Gabidulin Codes
We present a new family of maximum rank distance (MRD) codes. The new class
contains codes that are neither equivalent to a generalised Gabidulin nor to a
twisted Gabidulin code, the only two known general constructions of linear MRD
codes.Comment: 10 pages, accepted at the International Workshop on Coding and
Cryptography (WCC) 201
An extension of Overbeck's attack with an application to cryptanalysis of Twisted Gabidulin-based schemes
In the present article, we discuss the decoding of Gabidulin and related
codes from a cryptographic perspective and we observe that these codes can be
decoded with the single knowledge of a generator matrix. Then, we extend and
revisit Gibson's and Overbeck's attacks on the generalised GPT encryption
scheme (instantiated with Gabidulin codes) for various ranks of the distortion
matrix and apply our attack to the case of an instantiation with twisted
Gabidulin codes
Decoding and constructions of codes in rank and Hamming metric
As coding theory plays an important role in data transmission, decoding algorithms for new families of error correction codes are of great interest. This dissertation is dedicated to the decoding algorithms for new families of maximum rank distance (MRD) codes including additive generalized twisted Gabidulin (AGTG) codes and Trombetti-Zhou (TZ) codes, decoding algorithm for Gabidulin codes beyond half the minimum distance and also encoding and decoding algorithms for some new optimal rank metric codes with restrictions.
We propose an interpolation-based decoding algorithm to decode AGTG codes where the decoding problem is reduced to the problem of solving a projective polynomial equation of the form q(x) = xqu+1 +bx+a = 0 for a,b ∈ Fqm. We investigate the zeros of q(x) when gcd(u,m)=1 and proposed a deterministic algorithm to solve a linearized polynomial equation which has a close connection to the zeros of q(x).
An efficient polynomial-time decoding algorithm is proposed for TZ codes. The interpolation-based decoding approach transforms the decoding problem of TZ codes to the problem of solving a quadratic polynomial equation. Two new communication models are defined and using our models we manage to decode Gabidulin codes beyond half the minimum distance by one unit. Our models also allow us to improve the complexity for decoding GTG and AGTG codes.
Besides working on MRD codes, we also work on restricted optimal rank metric codes including symmetric, alternating and Hermitian rank metric codes. Both encoding and decoding algorithms for these optimal families are proposed. In all the decoding algorithms presented in this thesis, the properties of Dickson matrix and the BM algorithm play crucial roles.
We also touch two problems in Hamming metric. For the first problem, some cryptographic properties of Welch permutation polynomial are investigated and we use these properties to determine the weight distribution of a binary linear codes with few weights. For the second one, we introduce two new subfamilies for maximum weight spectrum codes with respect to their weight distribution and then we investigate their properties.Doktorgradsavhandlin
The -extended twisted generalized Reed-Solomon code
In this paper, we give a parity check matrix for the -extended twisted
generalized Reed Solomon (in short, ETGRS) code, and then not only prove that
it is MDS or NMDS, but also determine the weight distribution. Especially,
based on Schur method, we show that the -ETGRS code is not GRS or EGRS.
Furthermore, we present a sufficient and necessary condition for any punctured
code of the -ETGRS code to be self-orthogonal, and then construct several
classes of self-dual -TGRS codes and almost self-dual -ETGRS codes
Chaves mais pequenas para criptossistemas de McEliece usando codificadores convolucionais
The arrival of the quantum computing era is a real threat to the confidentiality
and integrity of digital communications. So, it is urgent to develop alternative
cryptographic techniques that are resilient to quantum computing. This is the
goal of pos-quantum cryptography. The code-based cryptosystem called
Classical McEliece Cryptosystem remains one of the most promising postquantum
alternatives. However, the main drawback of this system is that the
public key is much larger than in the other alternatives. In this thesis we study
the algebraic properties of this type of cryptosystems and present a new variant
that uses a convolutional encoder to mask the so-called Generalized Reed-
Solomon code. We conduct a cryptanalysis of this new variant to show that
high levels of security can be achieved using significant smaller keys than in
the existing variants of the McEliece scheme. We illustrate the advantages of
the proposed cryptosystem by presenting several practical examples.A chegada da era da computação quântica é uma ameaça real à
confidencialidade e integridade das comunicações digitais. É, por isso, urgente
desenvolver técnicas criptográficas alternativas que sejam resilientes à
computação quântica. Este é o objetivo da criptografia pós-quântica. O
Criptossistema de McEliece continua a ser uma das alternativas pós-quânticas
mais promissora, contudo, a sua principal desvantagem é o tamanho da chave
pública, uma vez que é muito maior do que o das outras alternativas. Nesta
tese estudamos as propriedades algébricas deste tipo de criptossistemas e
apresentamos uma nova variante que usa um codificador convolucional para
mascarar o código de Generalized Reed-Solomon. Conduzimos uma
criptoanálise dessa nova variante para mostrar que altos níveis de segurança
podem ser alcançados usando uma chave significativamente menor do que as
variantes existentes do esquema de McEliece. Ilustramos, assim, as vantagens
do criptossistema proposto apresentando vários exemplos práticos.Programa Doutoral em Matemátic
An extension of Overbeck\u27s attack with an application to cryptanalysis of Twisted Gabidulin-based schemes.
In this article, we discuss the decoding of Gabidulin and related codes from a cryptographic point of view, and we observe that these codes can be decoded solely from the knowledge of a generator matrix. We then extend and revisit Gibson and Overbeck attacks on the generalized GPT encryption scheme (instantiated with the Gabidulin code) for different ranks of the distortion matrix. We apply our attack to the case of an instantiation with twisted Gabidulin codes
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