2,970 research outputs found
Quantum codes from a new construction of self-orthogonal algebraic geometry codes
[EN] We present new quantum codes with good parameters which are constructed from self-orthogonal algebraic geometry codes. Our method permits a wide class of curves to be used in the formation of these codes. These results demonstrate that there is a lot more scope for constructing self-orthogonal AG codes than was previously known.G. McGuire was partially supported by Science Foundation Ireland Grant 13/IA/1914. The remainder authors were partially supported by the Spanish Government and the EU funding program FEDER, Grants MTM2015-65764-C3-2-P and PGC2018-096446-B-C22. F. Hernando and J. J. Moyano-Fernandez are also partially supported by Universitat Jaume I, Grant UJI-B2018-10.Hernando, F.; Mcguire, G.; Monserrat Delpalillo, FJ.; Moyano-Fernández, JJ. (2020). Quantum codes from a new construction of self-orthogonal algebraic geometry codes. 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Cryptanalysis of McEliece Cryptosystem Based on Algebraic Geometry Codes and their subcodes
We give polynomial time attacks on the McEliece public key cryptosystem based
either on algebraic geometry (AG) codes or on small codimensional subcodes of
AG codes. These attacks consist in the blind reconstruction either of an Error
Correcting Pair (ECP), or an Error Correcting Array (ECA) from the single data
of an arbitrary generator matrix of a code. An ECP provides a decoding
algorithm that corrects up to errors, where denotes
the designed distance and denotes the genus of the corresponding curve,
while with an ECA the decoding algorithm corrects up to
errors. Roughly speaking, for a public code of length over ,
these attacks run in operations in for the
reconstruction of an ECP and operations for the reconstruction of an
ECA. A probabilistic shortcut allows to reduce the complexities respectively to
and . Compared to the
previous known attack due to Faure and Minder, our attack is efficient on codes
from curves of arbitrary genus. Furthermore, we investigate how far these
methods apply to subcodes of AG codes.Comment: A part of the material of this article has been published at the
conferences ISIT 2014 with title "A polynomial time attack against AG code
based PKC" and 4ICMCTA with title "Crypt. of PKC that use subcodes of AG
codes". This long version includes detailed proofs and new results: the
proceedings articles only considered the reconstruction of ECP while we
discuss here the reconstruction of EC
Algebraic geometric construction of a quantum stabilizer code
The stabilizer code is the most general algebraic construction of quantum
error-correcting codes proposed so far. A stabilizer code can be constructed
from a self-orthogonal subspace of a symplectic space over a finite field. We
propose a construction method of such a self-orthogonal space using an
algebraic curve. By using the proposed method we construct an asymptotically
good sequence of binary stabilizer codes. As a byproduct we improve the
Ashikhmin-Litsyn-Tsfasman bound of quantum codes. The main results in this
paper can be understood without knowledge of quantum mechanics.Comment: LaTeX2e, 12 pages, 1 color figure. A decoding method was added and
several typographical errors were corrected in version 2. The description of
the decoding problem was completely wrong in version 1. In version 1 and 2,
there was a critical miscalculation in the estimation of parameters of codes,
and the constructed sequence of codes turned out to be worse than existing
ones. The asymptotically best sequence of quantum codes was added in version
3. Section 3.2 appeared in IEEE Transactions on Information Theory, vol. 48,
no. 7, pp. 2122-2124, July 200
Steane-Enlargement of Quantum Codes from the Hermitian Curve
In this paper, we study the construction of quantum codes by applying
Steane-enlargement to codes from the Hermitian curve. We cover
Steane-enlargement of both usual one-point Hermitian codes and of order bound
improved Hermitian codes. In particular, the paper contains two constructions
of quantum codes whose parameters are described by explicit formulae, and we
show that these codes compare favourably to existing, comparable constructions
in the literature.Comment: 11 page
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
The Road From Classical to Quantum Codes: A Hashing Bound Approaching Design Procedure
Powerful Quantum Error Correction Codes (QECCs) are required for stabilizing
and protecting fragile qubits against the undesirable effects of quantum
decoherence. Similar to classical codes, hashing bound approaching QECCs may be
designed by exploiting a concatenated code structure, which invokes iterative
decoding. Therefore, in this paper we provide an extensive step-by-step
tutorial for designing EXtrinsic Information Transfer (EXIT) chart aided
concatenated quantum codes based on the underlying quantum-to-classical
isomorphism. These design lessons are then exemplified in the context of our
proposed Quantum Irregular Convolutional Code (QIRCC), which constitutes the
outer component of a concatenated quantum code. The proposed QIRCC can be
dynamically adapted to match any given inner code using EXIT charts, hence
achieving a performance close to the hashing bound. It is demonstrated that our
QIRCC-based optimized design is capable of operating within 0.4 dB of the noise
limit
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