RT info:eu-repo/semantics/article
T1 Computational Aspects of Retrieving a Representation of an Algebraic Geometry Code
A1 Márquez Corbella, Irene
A1 Martínez Moro, Edgar
A1 Pellikaan, Ruud
A1 Ruano Benito, Diego
AB Code-based cryptography is an interesting alternative to classic number-theoretic public key cryptosystem since it is conjectured to be secure against quantum computer attacks. Many families of codes have been proposed for these cryptosystems such as algebraic geometry codes. In [Designs, Codes and Cryptography, pages 1-16, 2012] -for so called very strong algebraic geometry codes $\mathcal C=C_L(\mathcal X, \mathcal P, E)$, where $\mathcal X$ is an algebraic curve over $\mathbb F_q$, $\mathcal P$ is an $n$-tuple of mutually distinct $\mathbb F_q$-rational points of $\mathcal X$ and $E$ is a divisor of $\mathcal X$ with disjoint support from $\mathcal P$ --- it was shown that an equivalent representation $\mathcal C=C_L(\mathcal Y, \mathcal Q, F)$ can be found. The $n$-tuple of points is obtained directly from a generator matrix of $\mathcal C$, where the columns are viewed as homogeneous coordinates of these points. The curve $\mathcal Y$ is given by $I_2(\mathcal Y)$, the homogeneous elements of degree $2$ of the vanishing ideal $I(\mathcal Y)$. Furthermore, it was shown that $I_2(\mathcal Y)$ can be computed efficiently as the kernel of certain linear map. What was not shown was how to get the divisor $F$ and how to obtain efficiently an adequate decoding algorithm for the new representation. The main result of this paper is an efficient computational approach to the first problem, that is getting $F$. The security status of the McEliece public key cryptosystem using algebraic geometry codes is still not completely settled and is left as an open problem
YR 2014
FD 2014
LK http://uvadoc.uva.es/handle/10324/31738
UL http://uvadoc.uva.es/handle/10324/31738
LA eng
NO Journal of Symbolic Computation. Volume 64, pages 67-87 (2014)
NO Producción Científica
DS UVaDOC
RD 08-nov-2024