High Order Elements in Finite Fields Arising from Recursive Towers

TL;DR

This paper introduces recursive methods to construct towers of finite fields that generate elements of high order, with improved numerical bounds, for applications in finite field theory and cryptography.

Contribution

It presents new recursive constructions for field towers that produce high order elements, with better numerical bounds than previous methods.

Findings

Constructed towers over GF(q) with high order elements

Numerical analysis shows improved bounds on element orders

Examples demonstrate the effectiveness of the recursive approach

Abstract

We provide a recipe to construct towers of fields producing high order elements in $\mathrm{GF}(q,2^n)$, for odd $q$, and in $\mathrm{GF}(2,2 \cdot 3^n)$, for $n \ge 1$. These towers are obtained recursively by $x_{n}^2 + x_{n} = v(x_{n - 1})$, for odd $q$, or $x_{n}^3 + x_{n} = v(x_{n - 1})$, for $q=2$, where $v(x)$ is a polynomial of small degree over the prime field $\mathrm{GF}(q,1)$ and $x_n$ belongs to the finite field extension $\mathrm{GF}(q,2^n)$, for $q$ odd, or to $\mathrm{GF}(2,2\cdot 3^n)$. Several examples are carried out and analysed numerically. The lower bounds of the orders of the groups generated by $x_n$, or by the discriminant $\delta_n$ of the polynomial, are similar to the ones obtained in [BCG+09], but we get better numerical results in some cases.