The Journal of Korean Institute of Communications and Information Sciences (한국통신학회논문지)

The Korean Institute of Commucations and Information Sciences (한국통신학회)
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Two Types of Algorithms for Finding the Cube Root in Finite Fields

유한체상에서 세제곱근을 찾는 두 종류의 알고리즘

  • Cho, Gook Hwa (Ewha Womans University, Institute of Mathematical Sciences)
  • Received : 2016.02.25
  • Accepted : 2016.04.27
  • Published : 2016.05.31
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Abstract

We study algorithms that can efficiently find cube roots by modifying Cipolla-Lehmer algorithm. In this paper, we present two type algorithms for finding cube roots in finite field, which improves Cipolla-Lehmer algorithm. If the number of multiplications of two type algorithms has a little bit of a difference, then it is more efficient algorithm which have less storage variables.

Cipolla-Lehmer 알고리즘을 향상시킨 새로운 알고리즘을 통해 효율적으로 세제곱근을 찾을 수 있는 방법을 연구하였다. 본 논문에서는 일반적인 Cipolla-Lehmer 알고리즘보다 곱셈량을 줄인 향상된 두 가지 알고리즘을 소개한다. 유한체상에서 세제곱근을 찾는 곱셈량이 비슷한 두 가지 알고리즘을 제안하고, 곱셈량이 비슷하더라도 저장변수의 개수가 적을수록 효율적임을 보인다.

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References

  1. G. H. Cho, N. Koo, E. Ha, and S. Kwon, "New cube root algorithm based on the third order linear recurrence relations in finite fields," Designs, Codes and Cryptography, vol. 75, no. 3, pp. 483-495, 2015. https://doi.org/10.1007/s10623-013-9910-8
  2. M. Cipolla, "Un metodo per la risoluzione della congruenza di secondo grado," Rendicono dell'Accademia Scienze Fisiche e Matematiche, vol. 9, no. 3, pp. 154-163, 1903.
  3. L. E. Dickson, "Criteria for the irreducibility of functions in a finite field," Bull. Am. Math. Soc., vol. 13, no. 1, pp. 1-8, 1906. https://doi.org/10.1090/S0002-9904-1906-01403-3
  4. G. Gong, L. Harn, and H. Wu, "The GH public-key cryptosystem," Selected Areas in Cryptography, Springer Berlin Heidelberg, pp. 284-300, Dec. 2001.
  5. D. H. Lehmer, "Computer technology applied to the theory of numbers," Studies in Number Theory, Math. Assoc. Am. (distributed by Prentice-Hall, Englewood Cliffs, N. J.), pp. 117-151, 1969.
  6. N. Nishihara, R. Harasawa, Y. Sueyoshi, and A. Kudo, A remark on the computation of cube roots in finite fields, preprint, 2009, from http://eprint.iacr.org/2009/457.pdf.
  7. NIST, Digital Signature Standard, Federal Information Processing Standard 186-3, 2000, from http://csrc.nist.gov/publications/fips/.
  8. D. Shanks, "Five number-theoretic algorithms," in Proc. Second Manitoba Conf. Numerical Math., pp. 51-70, Winnipeg, Canada, Oct. 1972.
  9. A. Tonelli, "Bemerkung uber die Auflosung Quadratisher Congruenzen," Gottinger Nachrichten, pp. 344-346, 1891.