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Public key cryptosystems based on algebraic geometry codes


Ruud Pellikaan

Professor/a organitzador/a

Maria Bras-Amorós


Technische Universiteit Eindhoven


10-11-2014 12:00


This is a report on the McEliece public key cryptosystem based on algebraic geometry codes. In [3, 4] it is explained how a representation (Y, Q, F ) is retrieved efficiently from a generator matrix of the algebraic geometry code CL (X , P, E) for a given curve X , an n-tuple P of points on the curve and a divisor E. This result is not sufficient to obtain a polynomial attack on the public key cryptosystem based on these codes. In [1] a polynomial time attack is given using the idea of an error correcting pair [5]. It runs in O(n4 ) elementary operations in Fq , where n denotes the code length. Compared to previous attacks, it allows to recover a decoding algorithm for the public key even for codes from high genus curves. Recently the attack was extended to subcodes of algebraic geometry codes [2]. The problem for subfield subcodes of algebraic geometry codes is still open. Joint work with Alain Couvreur, Irene Márquez-Corbella, Edgar Martínez-Moro and Diego Ruano. See [1, 3, 4]. -------------------------------------------------- References [1] A. Couvreur and I. Marquez–Corbella and R. Pellikaan, “A polynomial time attack against algebraic geometry code based public key cryptosystems”, ISIT 2014, pp. 1446, 2014. [2] A. Couvreur and I. Marquez–Corbella and R. Pellikaan, “Cryptanalysis of public-key cryptosystems that use subcodes of algebraic geometry codes”, 4ICMCTA, 2014. [3] Márquez-Corbella, I. and Martínez-Moro, E. and Pellikaan, R. “On the unique representation of very strong algebraic geometry codes”, Designs, Codes and Cryptography 70, pp. 215-230, 2014. [4] Márquez-Corbella, I. and E. Martínez-Moro and R. Pellikaan and D. Ruano, “Computational aspects of retrieving a representation of an algebraic geometry code”, Journ. Symb. Comput. 64, pp. 67–87, 2014. [5] Pellikaan, R., “On decoding by error location and dependent sets of error positions”, Discrete Math. 106–107, pp. 369–381, 1992.


Lab 231