Paper 2013/041

Trace Expression of r-th Root over Finite Field

Gook Hwa Cho, Namhun Koo, 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 Müller'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 $\mathbb F_q$.

Metadata
Available format(s)
PDF
Category
Applications
Publication info
Published elsewhere. Unknown where it was published
Keywords
finite fieldr-th rootlinear recurrence relationTonelli-Shanks algorithmAdleman-Manders-Miller algorithmCipolla-Lehmer algorithm
Contact author(s)
shkwon7 @ gmail com
History
2013-01-30: revised
2013-01-29: received
See all versions
Short URL
https://ia.cr/2013/041
License
Creative Commons Attribution
CC BY

BibTeX

@misc{cryptoeprint:2013/041,
      author = {Gook Hwa Cho and Namhun Koo and Eunhye Ha and Soonhak Kwon},
      title = {Trace Expression of r-th Root over Finite Field},
      howpublished = {Cryptology {ePrint} Archive, Paper 2013/041},
      year = {2013},
      url = {https://eprint.iacr.org/2013/041}
}
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