International Association
for Cryptologic Research

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.