Introduction to algebraic geometry

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Table of contents

Introduction 
Résumé
1 Algebraic geometry codes 
1.1 Coding theory
1.2 Introduction to algebraic geometry
1.3 AG codes: definition and properties
2 Code-based cryptography 
2.1 McEliece encryption scheme
2.2 Information Set Decoding (ISD)
2.3 Key recovery problem
2.4 Algebraic cryptanalysis of McEliece schemes using alternant codes
3 McEliece scheme using quasi-cyclic alternant codes 
3.1 Quasi-cyclic alternant codes
3.2 Structural analysis of the invariant code
3.3 Security reduction to the key security of the invariant code
3.4 Proposition of a scheme: BIG QUAKE
4 A structural attack on a scheme using quasi-dyadic alternant codes 
4.1 Presentation of the NIST submission: DAGS
4.2 Schur products and conductors
4.3 Using the QD structure to compute a conductor
4.4 Description of the attack
4.5 Algorithm, work factor and implementation
5 Short McEliece keys from codes on curves with positive genus 
5.1 Construction of quasi-cyclic SSAG codes
5.2 Structure of the invariant code
5.3 QC codes from a cyclic cover of the projective line
5.4 The McEliece scheme with QC Hermitian codes
Bibliography