On the unique representation of very strong algebraic geometry codes

Abstract

This paper addresses the question of retrieving the triple ${(\mathcal X,\mathcal P, E)}$ from the algebraic geometry code ${\mathcal C = \mathcal C_L(\mathcal X, \mathcal P, E)}$, where ${\mathcal X}$ is an algebraic curve over the finite field ${\mathbb F_q, \,\mathcal P}$ is an n-tuple of ${\mathbb F_q}$-rational points on ${\mathcal X}$ and E is a divisor on ${\mathcal X}$. If ${\deg(E)\geq 2g+1}$ where g is the genus of ${\mathcal X}$, then there is an embedding of ${\mathcal X}$ onto ${\mathcal Y}$ in the projective space of the linear series of the divisor E. Moreover, if ${\deg(E)\geq 2g+2}$, then ${I(\mathcal Y)}$, the vanishing ideal of ${\mathcal Y}$, is generated by ${I_2(\mathcal Y)}$, the homogeneous elements of degree two in ${I(\mathcal Y)}$. If ${n >2 \deg(E)}$, then ${I_2(\mathcal Y)=I_2(\mathcal Q)}$, where ${\mathcal Q}$ is the image of ${\mathcal P}$ under the map from ${\mathcal X}$ to ${\mathcal Y}$. These three results imply that, if ${2g+2\leq m < \frac{1}{2}n}$, an AG representation ${(\mathcal Y, \mathcal Q, F)}$ of the code ${\mathcal C}$ can be obtained just using a generator matrix of ${\mathcal C}$ where ${\mathcal Y}$ is a normal curve in ${\mathbb{P}^{m-g}}$ which is the intersection of quadrics. This fact gives us some clues for breaking McEliece cryptosystem based on AG codes provided that we have an efficient procedure for computing and decoding the representation obtained.