Computational aspects of retrieving a representation of an algebraic geometry code

  • Authors:

  • Irene Márquez-Corbella;Edgar Martínez-Moro;Ruud Pellikaan;Diego Ruano

  • Affiliations:

  • Grace INRIA Saclay íle-de-France, Laboratoire d'Informatique (LIX) UMR 7161 X-CNRS, 1 rue Honoré d'Estienne d'Orves, Campus de l'ícole Polytechnique, 91120 Palaiseau, France;Institute of Mathematics, University of Valladolid, Castilla, Spain and Department of Mathematics and Statistics, Eastern Kentucky University, USA;Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands;Department of Mathematical Sciences, Aalborg University, 9220 Aalborg Øst, Denmark

  • Venue:
  • Journal of Symbolic Computation
  • Year:
  • 2014

Quantified Score

Hi-index 0.00

Visualization

Abstract

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 Marquez-Corbella et al. (2012) - for so called very strong algebraic geometry codes C=C"L(X,P,E), where X is an algebraic curve over F"q, P is an n-tuple of mutually distinct F"q-rational points of X and E is a divisor of X with disjoint support from P - it was shown that an equivalent representation C=C"L(Y,Q,F) can be found. The n-tuple of points is obtained directly from a generator matrix of C, where the columns are viewed as homogeneous coordinates of these points. The curve Y is given by I"2(Y), the homogeneous elements of degree 2 of the vanishing ideal I(Y). Furthermore, it was shown that I"2(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.