TRACE EXPRESSION OF r-TH ROOT OVER FINITE FIELD

Abstract

Efficient computation of r-th root in F-q has many applications in computational number theory and many other related areas. We present a new r-th root formula which generalizes Muller's result on square root, and which provides a possible improvement of the CipollaLehmer type algorithms for general case. More precisely, for given r-th power c is an element of F-q, we show that there exists alpha is an element of Fq(r)>such that Tr(alpha(Sigma i=0r-1 qi)-r/r2)(r) = c, where Tr (alpha) = alpha + alpha(q) + alpha(q2) + . . . + a(qr-1) and a is a root of certain irreducible polynomial of degree r over F-q.