On the Goppa morphism

TL;DR

This paper introduces the Goppa morphism, a geometric tool linking moduli spaces of level structures to Grassmannians, aiding cryptographic analysis and security considerations of Goppa codes.

Contribution

It defines the Goppa morphism from the moduli space of level structures to Grassmannians, enabling geometric analysis of Goppa codes and their cryptographic properties.

Findings

Identified equations defining the Goppa morphism's image.

Determined degree ranges to avoid for cryptographic security.

Extended results to convolutional Goppa codes.

Abstract

We investigate the geometric foundations of the space of geometric Goppa codes using the Tsfasman-Vladut H-construction. These codes are constructed from level structures, which extend the classical Goppa framework by incorporating invertible sheaves and their trivializations over rational points. A key contribution is the definition of the Goppa morphism, a map from the universal moduli space of level structures, denoted $LS_{g,n,d}$, to certain Grassmannian $\mathrm{Gr}(k,n)$. This morphism allows problems related to distinguishing attacks and key recovery in the context of Goppa Code-based Cryptography to be translated into a geometric language, addressing questions about the equations defining the image of the Goppa morphism and its fibers. Furthermore, we identify the ranges of the degree parameter $d$ that should be avoided to maintain security against distinguishers. Our results, valid over arbitrary base fields, also apply to convolutional Goppa codes.