**New Cube Root Algorithm Based on Third Order Linear Recurrence Relation in Finite Field**

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

**Abstract: **In this paper, we present a new cube root algorithm in finite
field $\mathbb{F}_{q}$ with $q$ a power of prime, which extends
the Cipolla-Lehmer type algorithms \cite{Cip,Leh}. Our cube root
method is inspired by the work of M\"{u}ller \cite{Muller} on
quadratic case. For given cubic residue $c \in \mathbb{F}_{q}$
with $q \equiv 1 \pmod{9}$, we show that there is an irreducible
polynomial $f(x)=x^{3}-ax^{2}+bx-1$ with root $\alpha \in
\mathbb{F}_{q^{3}}$ such that $Tr(\alpha^{\frac{q^{2}+q-2}{9}})$
is a cube root of $c$. Consequently we find an efficient cube root
algorithm based on third order linear recurrence sequence arising
from $f(x)$. Complexity estimation shows that our algorithm is
better than previously proposed Cipolla-Lehmer type algorithms.

**Category / Keywords: **applications / cube root algorithm, Cipolla-Lehmer algorithm

**Date: **received 16 Jan 2013

**Contact author: **shkwon at skku edu

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

**Version: **20130124:195739 (All versions of this report)

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