Canal-U

4.8. Attack against Algebraic Geometry codes

In this session, we will present an attack against Algebraic Geometry codes (AG codes). Algebraic Geometry codes is determined by a triple. First of all, an algebraic curve of genus g, then a n-tuple of rational points and then a divisor which has disjoint support from the n-tuple P. Then, the Algebraic Geometry code is obtained by evaluating at P all functions that belong to the vector space associated to the divisor E. Some properties of these codes are nearly optimal codes, that is, their designed minimum distance is nearly the optimal one. Moreover, the dual of an AG-code is again an AG-code. What about using Algebraic Geometry codes in code-based cryptography? Janwa and Moreno suggest to use Algebraic Geometry codes for the McEliece cryptosystem. This is a suitable proposal since these codes are nearly optimal and have efficient decoding algorithms. If we talk about codes over curves of genus zero then we are talking about generalized Reed-Solomon codes, as we will see in the next slides. So, for a curve of genus 0, this proposal is broken. If we talk about codes over curves of genus 1 and 2, then this proposal is broken by Faure and Minder. However, this attack has several drawbacks which makes it impossible to extend to a higher genera. But there is an attack for the general case. We will explain here this general attack. First over generalized Reed-Solomon codes and then we will give an idea on how it works for the general case. Recall that the generalized Reed-Solomon codes are Algebraic Geometry codes over curves of genus 0. Indeed, if we consider the projective line, this curve has genus 0 and its points are of the form (x:y) Now, we will consider P the n-tuple of points formed by these points and we take E to be K-1 times the point at the infinity. A basis of the vector space associated to this divisor is the following one. And if we evaluate this basis at the points P, we get a generator matrix of this AG code, which is also a generator matrix of a generalized Reed-Solomon code of dimension k associated to the pair (a,1), the all-ones vector.