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 \(c\in \mathbb {F}_q\) 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 \(\mathbb {F}_q\), and the algorithm of K. S. Williams and K. Hardy finds a solution in \(O(r^4+r^2\log q)\) multiplications in \(\mathbb {F}_q\). Our refinement finds a solution in \(O(r^3+r^2\log q)\) multiplications in \(\mathbb {F}_q\). 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.
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Acknowledgements
The authors would like to thank the anonymous reviewers for their valuable suggestions and kind comments. This research was supported by the National Research Foundation of Korea (KRF) Grant funded by the Korea government (MSIP) (No. 2016R1A5A1008055). The work of Soonhak Kwon was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2016R1D1A1B03931912).
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Cho, G.H., Go, B., Kim, C.H. et al. On the Cipolla–Lehmer type algorithms in finite fields. AAECC 30, 135–145 (2019). https://doi.org/10.1007/s00200-018-0362-2
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DOI: https://doi.org/10.1007/s00200-018-0362-2