Binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes

TL;DR

This paper characterizes certain irreducible polynomials related to Goppa codes and constructs binary irreducible quasi-cyclic parity-check subcodes, advancing cryptographic code design.

Contribution

It provides a necessary and sufficient condition for specific irreducible polynomials and constructs new quasi-cyclic subcodes of Goppa codes.

Findings

Characterization of irreducible polynomials of degrees 2s and 3s

Construction of binary irreducible quasi-cyclic parity-check subcodes

Enhanced understanding of Goppa code structures

Abstract

Goppa codes are particularly appealing for cryptographic applications. Every improvement of our knowledge of Goppa codes is of particular interest. In this paper, we present a sufficient and necessary condition for an irreducible monic polynomial $g(x)$ of degree $r$ over $\mathbb{F}_{q}$ satisfying $\gamma g(x)=(x+d)^rg({A}(x))$, where $q=2^n$, $A=\left(\begin{array}{cc} a&b\\1&d\end{array}\right)\in PGL_2(\Bbb F_{q})$, $\mathrm{ord}(A)$ is a prime, $g(a)\ne 0$, and $0\ne \gamma\in \Bbb F_q$. And we give a complete characterization of irreducible polynomials $g(x)$ of degree $2s$ or $3s$ as above, where $s$ is a positive integer. Moreover, we construct some binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes.