Details:
Title  Computational aspects of retrieving a representation of an algebraic geometry code  Author(s)  Irene MárquezCorbella, Edgar MartinezMoro, Ruud Pellikaan, Diego Ruano  Type  Article in Journal  Abstract  Abstract Codebased cryptography is an interesting alternative to classic numbertheoretic 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 MárquezCorbella 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 ntuple 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 ntuple 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.  Keywords  Gröbner basis  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S0747717113001740 
Language  English  Journal  Journal of Symbolic Computation  Volume  64  Number  0  Pages  67  87  Year  2014  Note  Mathematical and computer algebra techniques in cryptology  Edition  0  Translation 
No  Refereed 
No 
