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

The Korean Institute of Commucations and Information Sciences (한국통신학회)
권호 표지
DOI QR코드

DOI QR Code

Square Root Algorithm in Fq for Special Class of Finite Fields

특정한 유한체 Fq상에서의 제곱근 알고리즘

  • Received : 2012.12.26
  • Accepted : 2013.08.13
  • Published : 2013.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

We present a square root algorithm in $F_q$ which generalizes Atkin's square root algorithm [9] for finite field $F_q$ of q elements where $q{\equiv}5$ (mod 8) and Kong et al.'s algorithm [11] for the case $q{\equiv}9$ (mod 16). Our algorithm precomputes ${\xi}$ a primitive $2^s$-th root of unity where s is the largest positive integer satisfying $2^s|q-1$, and is applicable for the cases when s is small. The proposed algorithm requires one exponentiation for square root computation and is favorably compared with the algorithms of Atkin, M$\ddot{u}$ller and Kong et al.

$q{\equiv}5$ (mod 8)의 경우에 유한체 $F_q$상에서 Atkin의 제곱근 알고리즘과 $q{\equiv}9$ (mod 16)의 경우에 Kong의 알고리즘으로부터 일반적인 제곱근 알고리즘을 제안한다. 우리의 알고리즘은 s가 $2^s|q-1$을 만족하는 가장 큰 양의 정수라 할 때, $2^s$차 원시근 ${\xi}$를 미리 계산하였고 s의 값이 작을 때 적용가능하다. 제시한 알고리즘은 제곱근을 계산하기 위해 한 번의 지수계산이 필요하고, Akin, M$\ddot{u}$ller, Kong의 알고리즘과 비교해보아도 유리하다.

Keywords

References

  1. NIST, Digital Signature Standard, Federal Information Processing Standard 186-3, 2000, http://csrc.nist.gov/publications/fips/.
  2. D. Shanks, "Five number-theoretic algorithms," in Proc. Second Manitoba Conf. Numerical Math., pp. 51-70, Winnipeg, Canada, Oct. 1972.
  3. A. Tonelli, "Bemerkung uber die Auflosung Quadratisher Congruenzen," Gottinger Nachrichten, pp. 344-346, 1891.
  4. M. Cipolla, "Un metodo per la risoluzione della congruenza di secondo grado," Rendiconto dell'Accademia Scienze Fisiche e Matematiche, vol. 9, no. 3, pp. 154-163, 1903.
  5. D. H. Lehmer, "Computer technology applied to the theory of numbers," Studies in Number Theory, Math. Assoc. Amer. (distributed by Prentice-Hall, Englewood Cliffs, N.J.), pp. 117-151, 1969.
  6. S. Lindhurst, "An analysis of Shanks's algorithm for computing square roots in finite fields," CRM Proc. Lecture Notes, vol. 19, pp. 231-242, 1999.
  7. D. G. Han, D. Choi, and H. Kim, "Improved computation of square roots in specific finite fields," IEEE Trans. Comput., vol. 58, no. 2, pp. 188-196, Feb. 2009. https://doi.org/10.1109/TC.2008.201
  8. D.-G. Han, D. H. Choi, H. Kim, and J. Lim, "Efficient computation of square roots in finite fields $F_{{p}^{k}}$," J. Korea Inst. Inform. Security Cryptology (KIISC), vol. 18, no. 6A, pp. 3-15, Dec. 2008.
  9. A. O. L. Atkin, "Probabilistic primality testing," summary by F. Morain, Inria Research Report 1779, pp. 159-163, 1992.
  10. S. Muller, "On the computation of square roots in finite fields," Designs, Codes and Cryptography, vol. 31, no. 3, pp. 301-312, Mar. 2004. https://doi.org/10.1023/B:DESI.0000015890.44831.e2
  11. F. Kong, Z. Cai, J. Yu, and D. Li, "Improved generalized Atkin algorithm for computing square roots in finite fields," Inform. Process. Lett., vol. 98, no. 1, pp. 1-5, Apr. 2006. https://doi.org/10.1016/j.ipl.2005.11.015