**Trace Expression of r-th Root over Finite Field**

*Gook Hwa Cho and Namhun Koo and Eunhye Ha and Soonhak Kwon*

**Abstract: **Efficient computation of $r$-th root in $\mathbb F_q$ has many
applications in computational number theory and many other related
areas. We present a new $r$-th root formula which generalizes
M\"{u}ller's result on square root, and which provides a possible
improvement of the Cipolla-Lehmer algorithm for general case. More
precisely, for given $r$-th power $c\in \mathbb F_q$, we show that
there exists $\alpha \in \mathbb F_{q^r}$ such that
$Tr\left(\alpha^\frac{(\sum_{i=0}^{r-1}q^i)-r}{r^2}\right)^r=c$
where $Tr(\alpha)=\alpha+\alpha^q+\alpha^{q^2}+\cdots
+\alpha^{q^{r-1}}$ and $\alpha$ is a root of certain irreducible
polynomial of degree $r$ over $\mathbb F_q$.

**Category / Keywords: **applications / finite field, r-th root, linear recurrence relation, Tonelli-Shanks algorithm, Adleman-Manders-Miller algorithm, Cipolla-Lehmer algorithm

**Date: **received 26 Jan 2013, last revised 30 Jan 2013

**Contact author: **shkwon7 at gmail com

**Available format(s): **PDF | BibTeX Citation

**Version: **20130130:100027 (All versions of this report)

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