On Taking r-th Roots without r-th Nonresidues over Finite Fields and Its Applications

Abstract

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.