论文标题
正常碱基的次级时间算法
Subquadratic-Time Algorithms for Normal Bases
论文作者
论文摘要
对于任何有限的Galois字段扩展$ \ Mathsf {K}/\ Mathsf {f} $，带有Galois组$ G = \ Mathrm {gal}（\ Mathsf {K}/\ Mathsf {k}/\ Mathsf {f}）$ $ \ mathsf {f} $ - $ \ mathsf {k} $的基础。这样的$α$称为普通元素，$ g \cdotα$是正常的基础。我们引入了一种用于测试\ Mathsf {K} $的概率算法是正常的，当$ G $是有限的Abelian还是MetacyClic组时。该算法是基于以下事实：确定$α$是否正常可以降低为确定$ \ sum_ {g \ in g} g} g（α）g \ in \ mathsf {k} [k} [g] $是可逆的；它需要稍微二等的操作数量。一旦我们知道$α$是正常的，我们就会展示如何在$ \ mathsf {k}/\ mathsf {f} $的功率基础之间执行转换，并以相同的渐近成本进行正常基础。
For any finite Galois field extension $\mathsf{K}/\mathsf{F}$, with Galois group $G = \mathrm{Gal}(\mathsf{K}/\mathsf{F})$, there exists an element $α\in \mathsf{K}$ whose orbit $G\cdotα$ forms an $\mathsf{F}$-basis of $\mathsf{K}$. Such a $α$ is called a normal element and $G\cdotα$ is a normal basis. We introduce a probabilistic algorithm for testing whether a given $α\in \mathsf{K}$ is normal, when $G$ is either a finite abelian or a metacyclic group. The algorithm is based on the fact that deciding whether $α$ is normal can be reduced to deciding whether $\sum_{g \in G} g(α)g \in \mathsf{K}[G]$ is invertible; it requires a slightly subquadratic number of operations. Once we know that $α$ is normal, we show how to perform conversions between the power basis of $\mathsf{K}/\mathsf{F}$ and the normal basis with the same asymptotic cost.
