Browse > Article
http://dx.doi.org/10.13089/JKIISC.2008.18.6A.3

Efficient Computation of Square Roots in Finite Fields $F{_p}{^{k}}$  

Han, Dong-Guk (ETRI)
Choi, Doo-Ho (ETRI)
Kim, Ho-Won (Pusan National University)
Lim, Jong-In (Korea University)
Publication Information
Journal of the Korea Institute of Information Security & Cryptology / v.18, no.6A, 2008 , pp. 3-15 More about this Journal
Abstract
In this paper we study exponentiation in finite fields $F{_p}{^{k}}$(k is odd) with very special exponents such as they occur in algorithms for computing square roots. Our algorithmic approach improves the corresponding exponentiation independent of the characteristic of $F{_p}{^{k}}$. To the best of our knowledge, it is the first major improvement to the Tonelli-Shanks algorithm, for example, the number of multiplications can be reduced to at least 60% on average when $p{\equiv}1$ (mod 16). Several numerical examples are given that show the speed-up of the proposed methods.
Keywords
square roots; finite fields; Tonelli-Shanks algorithm;
Citations & Related Records
연도 인용수 순위
  • Reference
1 D. Shanks, 'Five number-theoretic algorithms,' Proceedings of the second Manitoba conference on numerical mathematics, pp.51-70, 1972
2 A. Tonelli, 'Bemerkung uber die Auflosung quadratischer Congruenzen,' Gottinger Nachrichten, pp. 344-346, 1891
3 W. Feng, Y. Nogami, Y. Morikawa, 'A fast square root computation using the Frobenius mapping,' ICICS 2003, LNCS 2836, pp. 1-10
4 M.K. Lee, H. Kim, D. Hong, K. Chung, 'Efficient exponentiation in GF($P^{m}$ ) using the Frobenius map,' ICCSA 2006 (Part 4), LNCS 3983, pp. 584-593
5 R. Lidl and H. Niederreiter, 'Finite Field,' Encyclopedia of Mathematics and Its Applications, vol. 20, Cambridge University Press, Cambridge, 1997
6 T. Itoh and S. Tsujii, 'A fast algorithm for computing multiplicative inverses in GF($2^{m}$ using normal bases,' Information and Computation, vol. 78, pp. 171-177, 1988   DOI
7 A.O.L. Atkin, 'Probabilistic primality testing,' Summary by F. Morain, INRIA Res. Rep. 1779, pp. 159-163, 1992.
8 P.S.L.M. Barreto and J.F. Voloch, 'Efficient Computation of Roots in Fintie Fields,' Journal of Design, Codes and Cryptography, vol. 39, pp. 275-280, 2006   DOI
9 F. Kong, Z. Cai, J. Yu, and D. Li, 'Improved generalized Atkin algorithm for computing square roots in finite fields,' Information processing Letters, vol. 98 (1), pp. 1-5, 2006   DOI   ScienceOn
10 V.S. Miller, 'Use of elliptic curves in cryptography,' In Advances in Cryptology-CRYPTO' 85, LNCS 218, pp. 417-426, 1986
11 S. Lindhurst, 'An analysis of Shanks's algorithm for computing square roots in finite fields,' CRM Proceedings and Lecture Notes, vol. 19, pp. 231-242, 1999
12 R. Crandall and C. Pomerance, 'Prime Numbers,' A Computational Perspective, Springer-Verlag, New York, 2001
13 S. Muller, 'On the computation of square roots in finite fields,' Journal of Design, Codes and Cryptography, vol. 31, pp. 301-312, 2004   DOI
14 N. Koblitz, 'Elliptic curve cryptosystems,' In Mathematics of Computation, volume 48, pp. 203-209, 1987   DOI
15 P.S.L.M. Barreto, H. Kim, B. Lynn, and M. Scott, 'Efficient Algorithms for Pairing-Based Cryptosystems,' CRYPTO 2002, LNCS 2442, pp. 354-368, 2002