论文标题
由递归塔产生的有限田中的高级元素
High Order Elements in Finite Fields Arising from Recursive Towers
论文作者
论文摘要
我们提供了一个配方，以构建塔的塔楼，该塔以$ \ mathrm {gf}（q，2^n）$，用于奇数$ q $，在$ \ mathrm {gf}（2,2 \ cdot 3^n）$中，以$ n \ ge 1 $。这些塔由$ x_ {n}^2 + x_ {n} = v（x_ {n -1}）$递归获得$ \ mathrm {gf}（q，1）$和$ x_n $属于有限字段扩展$ \ mathrm {gf}（q，2^n）$，对于$ q $奇数，或to $ q $ odd，或to $ \ mathrm {gf}（2,2 \ cdot 3^n）$。进行了几个示例并进行数值分析。由$ x_n $或多项式的判别$δ_n$生成的组的下限与[BCG+09]中获得的组相似，但是在某些情况下，我们获得了更好的数值结果。
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 $δ_n$ of the polynomial, are similar to the ones obtained in [BCG+09], but we get better numerical results in some cases.
