On the Menezes-Teske-Weng's conjecture

TL;DR

This paper proves the Menezes-Teske-Weng conjecture related to quadratic equations over binary extension fields, confirming its validity and enabling the complete characterization of the null space of certain linear polynomials.

Contribution

The paper provides a formal proof of the long-standing conjecture and applies it to determine the null space of a class of linear polynomials.

Findings

The conjecture is proven to be correct.

Complete characterization of the null space of specific linear polynomials.

Enhanced understanding of quadratic equations over binary extension fields.

Abstract

In 2003, Alfred Menezes, Edlyn Teske and Annegret Weng presented a conjecture on properties of the solutions of a type of quadratic equation over the binary extension fields, which had been convinced by extensive experiments but the proof was unknown until now. We prove that this conjecture is correct. Furthermore, using this proved conjecture, we have completely determined the null space of a class of linear polynomials.