High Order Elements in Finite Fields Arising from Recursive Towers

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.

References in corpus (1)

• Finite field elements of high order arising from modular curves