For any finite Galois field extension K/F, with Galois group G = Gal (K/F), there exists an element \(\alpha \in \) K whose orbit \(G\cdot\alpha\) forms an F-basis of K. Such an \(\alpha\) is called a normal element, and \(G\cdot\alpha\) is a normal basis. We introduce a probabilistic algorithm for testing whether a given \(\alpha \in\) K is normal, when G is either a finite abelian or a metacyclic group. The algorithm is based on the fact that deciding whether \(\alpha\) is normal can be reduced to deciding whether \(\sum_{g \in G} g(\alpha)g \in\) K[G] is invertible; it requires a slightly subquadratic number of operations. Once we know that \(\alpha\) is normal, we show how to perform conversions between the power basis of K/F and the normal basis with the same asymptotic cost.

对于任何有限Galois域扩展ķ/ F，与伽罗瓦组G ^ =半乳糖（K / F） ，存在一个元素\（\阿尔法\在\） ķ 其轨道\（G \ CDOT \阿尔法\）形成˚F - K的基础。这样的 \（\ alpha \）称为常规元素，而\（G \ cdot \ alpha \）是 常规基础。我们引入一种概率算法，以测试当给定的\（\ alpha \ in \） K是否正常时，G是一个有限阿贝尔群或一个元环群。该算法基于以下事实：\（\ alpha \）正常可以简化为\（\ sum_ {g \ in G} g（\ alphag g \ in \） K [ G ]是否可逆；它需要少量次次操作。一旦知道\（\ alpha \）是正常的，我们将说明如何以相同的渐近成本在K / F的幂基和正常基数之间执行转换。