31 research outputs found
Communication over Finite-Chain-Ring Matrix Channels
Though network coding is traditionally performed over finite fields, recent
work on nested-lattice-based network coding suggests that, by allowing network
coding over certain finite rings, more efficient physical-layer network coding
schemes can be constructed. This paper considers the problem of communication
over a finite-ring matrix channel , where is the channel
input, is the channel output, is random error, and and are
random transfer matrices. Tight capacity results are obtained and simple
polynomial-complexity capacity-achieving coding schemes are provided under the
assumption that is uniform over all full-rank matrices and is uniform
over all rank- matrices, extending the work of Silva, Kschischang and
K\"{o}tter (2010), who handled the case of finite fields. This extension is
based on several new results, which may be of independent interest, that
generalize concepts and methods from matrices over finite fields to matrices
over finite chain rings.Comment: Submitted to IEEE Transactions on Information Theory, April 2013.
Revised version submitted in Feb. 2014. Final version submitted in June 201
On the ideal associated to a linear code
This article aims to explore the bridge between the algebraic structure of a
linear code and the complete decoding process. To this end, we associate a
specific binomial ideal to an arbitrary linear code. The
binomials involved in the reduced Gr\"obner basis of such an ideal relative to
a degree-compatible ordering induce a uniquely defined test-set for the code,
and this allows the description of a Hamming metric decoding procedure.
Moreover, the binomials involved in the Graver basis of
provide a universal test-set which turns out to be a set containing the set of
codewords of minimal support of the code
Quasi-Cyclic Codes
Quasi-cyclic codes form an important class of algebraic codes that includes
cyclic codes as a special subclass. This chapter focuses on the algebraic
structure of quasi-cyclic codes, first. Based on these structural properties,
some asymptotic results, a few minimum distance bounds and further applications
such as the trace representation and characterization of certain subfamilies of
quasi-cyclic codes are elaborated. This survey will appear as a chapter in "A
Concise Encyclopedia of Coding Theory" to be published by CRC Press.Comment: arXiv admin note: text overlap with arXiv:1906.0496
A Combinatorial Commutative Algebra Approach to Complete Decoding
Esta tesis pretende explorar el nexo de unión que existe entre la estructura algebraica de un código lineal y el proceso de descodificación completa. Sabemos que el proceso de descodificación completa para códigos lineales arbitrarios es NP-completo, incluso si se admite preprocesamiento de los datos. Nuestro objetivo es realizar un análisis algebraico del proceso de la descodificación, para ello asociamos diferentes estructuras matemáticas a ciertas familias de códigos. Desde el punto de vista computacional, nuestra descripción no proporciona un algoritmo eficiente pues nos enfrentamos a un problema de naturaleza NP. Sin embargo, proponemos algoritmos alternativos y nuevas técnicas que permiten relajar las condiciones del problema reduciendo los recursos de espacio y tiempo necesarios para manejar dicha estructura algebraica.Departamento de Algebra, GeometrÃa y TopologÃ
Quantum stabilizer codes and beyond
The importance of quantum error correction in paving the way to build a
practical quantum computer is no longer in doubt. This dissertation makes a
threefold contribution to the mathematical theory of quantum error-correcting
codes. Firstly, it extends the framework of an important class of quantum codes
-- nonbinary stabilizer codes. It clarifies the connections of stabilizer codes
to classical codes over quadratic extension fields, provides many new
constructions of quantum codes, and develops further the theory of optimal
quantum codes and punctured quantum codes. Secondly, it contributes to the
theory of operator quantum error correcting codes also called as subsystem
codes. These codes are expected to have efficient error recovery schemes than
stabilizer codes. This dissertation develops a framework for study and analysis
of subsystem codes using character theoretic methods. In particular, this work
establishes a close link between subsystem codes and classical codes showing
that the subsystem codes can be constructed from arbitrary classical codes.
Thirdly, it seeks to exploit the knowledge of noise to design efficient quantum
codes and considers more realistic channels than the commonly studied
depolarizing channel. It gives systematic constructions of asymmetric quantum
stabilizer codes that exploit the asymmetry of errors in certain quantum
channels.Comment: Ph.D. Dissertation, Texas A&M University, 200
Practical Lattice Cryptosystems: NTRUEncrypt and NTRUMLS
Public key cryptography, as deployed on the internet today, stands on shaky
ground. For over twenty years now it has been known that the systems in
widespread use are insecure against adversaries equipped with quantum computers
-- a fact that has largely been discounted due to the enormous challenge of
building such devices. However, research into the development of quantum
computers is accelerating and is producing an abundance of positive results
that indicate quantum computers could be built in the near future. As a
result, individuals, corporations and government entities are calling for the deployment of
new cryptography to replace systems that are vulnerable to quantum
cryptanalysis. Few satisfying schemes are to be found.
This work examines the design, parameter selection, and cryptanalysis of a
post-quantum public key encryption scheme, NTRUEncrypt, and a related
signature scheme, NTRUMLS. It is hoped that this analysis will prove useful in
comparing these schemes against other candidates that have been proposed to
replace existing infrastructure
Usability of structured lattices for a post-quantum cryptography: practical computations, and a study of some real Kummer extensions
Lattice-based cryptography is an excellent candidate for post-quantum cryptography, i.e. cryptosystems which are resistant to attacks run on quantum computers. For efficiency reason, most of the constructions explored nowadays are based on structured lattices, such as module lattices or ideal lattices. The security of most constructions can be related to the hardness of retrieving a short element in such lattices, and one does not know yet to what extent these additional structures weaken the cryptosystems. A related problem – which is an extension of a classical problem in computational number theory – called the Short Principal Ideal Problem (or SPIP), consists of finding a short generator of a principal ideal. Its assumed hardness has been used to build some cryptographic schemes. However it has been shown to be solvable in quantum polynomial time over cyclotomic fields, through an attack which uses the Log-unit lattice of the field considered. Later, practical results showed that multiquadratic fields were also weak to this strategy.
The main general question that we study in this thesis is To what extent can structured lattices be used to build a post-quantum cryptography
Space programs summary no. 37-51, volume 3 for the period April 1 to May 31, 1968. Supporting research and advanced development
Space Programs Summary - supporting research and advanced developmen