Cryptanalysis of public-key cryptosystems based on algebraic geometry codes

This paper addresses the question of retrieving the triple (X ;P; E) from the algebraic geometry code CL(X ;P; E), where X is an algebraic curve over the finite field Fq, P is an n-tuple of Fq-rational points on X and E is a divisor on X . If deg(E) = 2g + 1 where g is the genus of X , then there is an embedding of X onto Y in the projective space of the linear series of the divisor E. Moreover, if deg(E) = 2g + 2, then I(Y), the vanishing ideal of Y, is generated by I2(Y), the homogeneous elements of degree two in I(Y). If n > 2 deg(E), then I2(Y) = I2(Q), where Q is the image of P under the map from X to Y. These two results imply that certain algebraic geometry codes are not secure if used in the McEliece public-key cryptosystem.