Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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TRACE EXPRESSION OF r-TH ROOT OVER FINITE FIELD

  • Cho, Gook Hwa (Institute of Mathematical Sciences Ewha Womans University) ;
  • Koo, Namhun (Institute of Mathematical Sciences Ewha Womans University) ;
  • Kwon, Soonhak (Department of Mathematics Sungkyunkwan University)
  • Received : 2019.08.19
  • Accepted : 2019.12.12
  • Published : 2020.07.01
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Abstract

Efficient computation of r-th root in 𝔽q 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 ∈ 𝔽q, we show that there exists α ∈ 𝔽qr 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 𝔽q.

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References

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