有限域上无r次非剩余的r次根及其应用
On Taking r-th Roots without r-th Nonresidues over Finite Fields and Its Applications
摘要：
我们首先显示了一个确定性的算法采取$r$ -th根在$\F_q$没有被给予任何$r$ -th非剩余，其中$\F_q$是一个有限域的$q$元素和$r$是一个小素数，使$r^2$除$q-1$。作为应用，我们给出了在$\F_q$上构造$r$次非剩余、构造本原元、求解多项式方程和计算椭圆曲线“$n$次根”的确定性算法，以及广义Proth数的确定性素性检验.所有的算法都是在不假设任何未经证明的假设的情况下证明的。只有当$q-1$上的所有因子都很小，并且在$\F_q$上可以有效地构造一些原单位根时，它们才是有效的。在某些情况下，它们是已知的确定性算法中最快的。
英文摘要：
We first show a deterministic algorithm for taking $r$-th roots over $\F_q$ without being given any $r$-th nonresidue, where $\F_q$ is a finite field with $q$ elements and $r$ is a small prime such that $r^2$ divides of $q-1$. As applications, we illustrate deterministic algorithms over $\F_q$ for constructing $r$-th nonresidues, constructing primitive elements, solving polynomial equations and computing elliptic curve "$n$-th roots", and a deterministic primality test for the generalized Proth numbers. All algorithms are proved without assuming any unproven hypothesis. They are efficient only if all the factors of $q-1$ are small and some primitive roots of unity can be constructed efficiently over $\F_q$. In some cases, they are the fastest among the known deterministic algorithms.
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