Skew differential Goppa codes and their application to McEliece cryptosystem

TL;DR

This paper introduces skew differential Goppa codes, extending classic Goppa codes to a non-commutative setting, and demonstrates their potential for constructing a McEliece-type cryptosystem with adjustable parameters.

Contribution

It defines a new class of non-commutative Goppa codes and develops an efficient decoding algorithm, enabling their application in cryptography.

Findings

Successful extension of Goppa codes to non-commutative algebraic structures

Development of an efficient decoding algorithm for these codes

Proposal of a McEliece-type cryptosystem using skew differential Goppa codes

Abstract

A class of linear codes that extends classic Goppa codes to a non-commutative context is defined. An efficient decoding algorithm, based on the solution of a non-commutative key equation, is designed. We show how the parameters of these codes, when the alphabet is a finite field, may be adjusted to propose a McEliece-type cryptosystem.