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On the Goppa morphism

Ángel Luis Muñoz Castañeda

TL;DR

The paper introduces a geometric framework for geometric Goppa codes via the Tsfasman-Vladut $H$-construction, recasting codes as images of evaluation maps from moduli of level structures. It constructs moduli stacks $\mathfrak LS_{g,n,d}$ and related categories, and defines the Goppa morphism to the Grassmannian Gr$(k,n)$ that associates to each level structure a code; under $n>d>2g-2$, the extended morphism is an immersion, embedding the level-structure space as a locally closed subvariety. It analyzes the duality theory, distinguishability, and self-duality, providing a framework to study security questions in code-based cryptography by translating them into algebraic geometry, with explicit genus-zero realizations as generalized Reed-Solomon codes and their duals. The results yield both structural insights (smoothness, DM-stacks, immersions) and practical consequences (ranges of $d$ to avoid for indistinguishability, genus-zero explicit formulas), and extend to convolutional Goppa codes, offering a versatile geometric lens for Goppa-code cryptography.

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.

On the Goppa morphism

TL;DR

The paper introduces a geometric framework for geometric Goppa codes via the Tsfasman-Vladut -construction, recasting codes as images of evaluation maps from moduli of level structures. It constructs moduli stacks and related categories, and defines the Goppa morphism to the Grassmannian Gr that associates to each level structure a code; under , the extended morphism is an immersion, embedding the level-structure space as a locally closed subvariety. It analyzes the duality theory, distinguishability, and self-duality, providing a framework to study security questions in code-based cryptography by translating them into algebraic geometry, with explicit genus-zero realizations as generalized Reed-Solomon codes and their duals. The results yield both structural insights (smoothness, DM-stacks, immersions) and practical consequences (ranges of to avoid for indistinguishability, genus-zero explicit formulas), and extend to convolutional Goppa codes, offering a versatile geometric lens for Goppa-code cryptography.

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 , to certain Grassmannian . 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 that should be avoided to maintain security against distinguishers. Our results, valid over arbitrary base fields, also apply to convolutional Goppa codes.

Paper Structure

This paper contains 29 sections, 19 theorems, 85 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Conventions and notations
  3. Preliminaries on Goppa codes
  4. Basic definitions
  5. Parameters
  6. Equivalent level structures
  7. A remark
  8. The dual code
  9. Space of level structures and Goppa morphism. Constructions
  10. Category of level structures $\mathfrak LS_{g,n,d}$ and Goppa functor
  11. The induced category $\mathfrak{LS}_{X,D,d}$ and Goppa functor
  12. Space of level structures and Goppa morphism. Geometry
  13. The category $\mathfrak LS_{X,D,d}$
  14. Structure
  15. Inmersions
  16. ...and 14 more sections

Key Result

Lemma 2.6

Let $\mathfrak{g}:=(X,p_1,\hdots,p_n,L,\gamma_1,\hdots,\gamma_{n})$ be a level structure of rank one.

Figures (1)

  • Figure 1: The general representation of the parabola $\Xi(d)$

Theorems & Definitions (52)

  • Lemma 2.6
  • proof
  • Definition 2.7
  • Lemma 2.9
  • proof
  • Remark 2.11
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Definition 3.4
  • ...and 42 more