Improvement Of Barreto-Voloch Algorithm For Computing $r$th Roots Over Finite Fields

Root extraction is a classical problem in computers algebra. It plays an essential role in cryptosystems based on elliptic curves. In 2006, Barreto and Voloch proposed an algorithm to compute $r$th roots in ${F}_{q^m}$ for certain choices of $m$ and $q$. If $r\,||\,q-1$ and $(m, r)=1,$ they proved that the complexity of their method is $\widetilde{\mathcal {O}}(r(\log m+\log\log q)m\log q)$. In this paper, we extend the Barreto-Voloch algorithm to the general case that $r\,||\,q^m-1$, without the restrictions $r\,||\,q-1$ and $(m, r)=1$. We also specify the conditions that the Barreto-Voloch algorithm can be preferably applied.