2024-06-18T08:42:01Zhttps://uvadoc.uva.es/oai/requestoai:uvadoc.uva.es:10324/317382021-06-24T07:41:07Zcom_10324_32197com_10324_952com_10324_894com_10324_1129com_10324_931col_10324_32199col_10324_1193
00925njm 22002777a 4500
dc
Márquez Corbella, Irene
author
Martínez Moro, Edgar
author
Pellikaan, Ruud
author
Ruano Benito, Diego
author
2014
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
Journal of Symbolic Computation. Volume 64, pages 67-87 (2014)
http://uvadoc.uva.es/handle/10324/31738
http://dx.doi.org/10.1016/j.jsc.2013.12.007
Computational Aspects of Retrieving a Representation of an Algebraic Geometry Code