• Bulletin of the Korean Mathematical Society
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• v.56 no.2
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• pp.277-283
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• 2019

In this paper, we present a cube root algorithm using a recurrence relation. Additionally, we compare the implementations of the Pocklington and $Padr{\acute{o}}-S{\acute{a}}ez$ algorithm with the Adleman-Manders-Miller algorithm. With the recurrence relations, we improve the Pocklington and $Padr{\acute{o}}-S{\acute{a}}ez$ algorithm by using a smaller base for exponentiation. Our method can reduce the average number of ${\mathbb{F}}_q$ multiplications.

• Journal of the Korean Mathematical Society
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• v.57 no.4
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• pp.1019-1030
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• 2020

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 Müller's result on square root, and which provides a possible improvement of the Cipolla-Lehmer type algorithms for general case. More precisely, for given r-th power c ∈ 𝔽_q, we show that there exists α ∈ 𝔽_q^r such that $$Tr{\left(\begin{array}{cccc}{{\alpha}^{{\frac{({\sum}_{i=0}^{r-1}\;q^i)-r}{r^2}}}\atop{\text{ }}}\end{array}\right)}^r=c,$$ where $Tr({\alpha})={\alpha}+{\alpha}^q+{\alpha}^{q^2}+{\cdots}+{\alpha}^{q^{r-1}}$ and α is a root of certain irreducible polynomial of degree r over 𝔽_q.