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On the Cipolla–Lehmer type algorithms in finite fields

Title
On the Cipolla–Lehmer type algorithms in finite fields
Authors
Cho G.H.Go B.Kim C.H.Koo N.Kwon S.
Ewha Authors
조국화
Issue Date
2019
Journal Title
Applicable Algebra in Engineering, Communications and Computing
ISSN
0938-1279JCR Link
Citation
Applicable Algebra in Engineering, Communications and Computing vol. 30, no. 2, pp. 135 - 145
Keywords
Adleman–Manders–Miller algorithmCipolla–Lehmer algorithmFinite fieldPrimitive rootr-th root
Publisher
Springer Verlag
Indexed
SCIE; SCOPUS WOS scopus
Document Type
Article
Abstract
In this paper, we present a refinement of the Cipolla–Lehmer type algorithm given by H. C. Williams in 1972, and later improved by K. S. Williams and K. Hardy in 1993. For a given r-th power residue c∈ F q where r is an odd prime, the algorithm of H. C. Williams determines a solution of X r = c in O(r 3 log q) multiplications in F q , and the algorithm of K. S. Williams and K. Hardy finds a solution in O(r 4 + r 2 log q) multiplications in F q . Our refinement finds a solution in O(r 3 + r 2 log q) multiplications in F q . Therefore our new method is better than the previously proposed algorithms independent of the size of r, and the implementation result via SageMath shows a substantial speed-up compared with the existing algorithms. It should be mentioned that our method also works for a composite r. © 2018, Springer-Verlag GmbH Germany, part of Springer Nature.
DOI
10.1007/s00200-018-0362-2
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연구기관 > 수리과학연구소 > Journal papers
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