Squares of bivariate Goppa codes

Abstract

In this paper, we study squares of bivariate Goppa codes, as they relate to the Goppa code distinguishing problem for bivariate Goppa codes. Introduced in 2021, multivariate Goppa codes are subfield subcodes of certain evaluation codes defined by evaluating polynomials in m variables. The evaluation codes are augmented Cartesian codes, a generalization of Reed-Muller codes. Classical Goppa codes are obtained by taking m=1. The multivariate Goppa code distinguishing problem is to distinguish efficiently a generator matrix of a multivariate Goppa code from a randomly drawn one. Because a randomly drawn code has a large square, codes with small squares may be considered distinguishable, revealing structure which facilitates private key recovery in a code-based cryptosystem.