Computational aspects of retrieving a representation of an algebraic geometry code

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 (Márquez-Corbella et al., 2012) — for so called very strong algebraic geometry codes C=CL(X,P,E), where X is an algebraic curve over Fq, P is an n-tuple of mutually distinct Fq-rational points of X and E is a divisor of X with disjoint support from P — it was shown that an equivalent representation C=CL(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 I2(Y), the homogeneous elements of degree 2 of the vanishing ideal I(Y). Furthermore, it was shown that I2(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: Public key cryptosystem; Code-based cryptography; Algebraic geometry codes; Gröbner basis