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 rth roots in Fqm for certain choices of m and q. If r || q − 1 and (m, r) = 1, they proved that the complexity of their method is Õ(r(logm + log log q)m log q). In this paper, we extend the Barreto-Voloch algorithm to the general case that r || q − 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.