Sequences of binary irreducible polynomials

TL;DR

This paper presents a method to generate an infinite sequence of binary irreducible polynomials starting from any initial irreducible polynomial, with degrees doubling after a finite initial segment, useful for applications in coding theory and cryptography.

Contribution

The paper introduces a novel construction for infinite binary irreducible polynomial sequences with predictable degree growth, starting from any initial irreducible polynomial.

Findings

Sequence of polynomials becomes degree-doubling after finite steps

Construction works for any initial irreducible polynomial over 

Finite initial segment length depends on the degree of the starting polynomial

Abstract

In this paper we construct an infinite sequence of binary irreducible polynomials starting from any irreducible polynomial $f_0 \in \F_2 [x]$. If $f_0$ is of degree $n = 2^l \cdot m$, where $m$ is odd and $l$ is a non-negative integer, after an initial finite sequence of polynomials $f_0, f_1, ..., f_{s}$ with $s \leq l+3$, the degree of $f_{i+1}$ is twice the degree of $f_i$ for any $i \geq s$.