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Improved Computation of Square Roots in Specific Finite Fields
February 2009 (vol. 58 no. 2)
pp. 188-196
Dong-Guk Han, Electronics and Telecommunications Research Institute, Daejeon
Dooho Choi, Electronics and Telecommunications Research Institute, Daejeon
Howon Kim, Pusan National University, Korea
In this paper, we study exponentiation in the specific finite fields {\bf F}_{q} with very special exponents such as those that occur in algorithms for computing square roots. Here, q is a prime power, q = p^{k}, where k> 1, and k is odd. Our algorithmic approach improves the corresponding exponentiation resulted from the better rewritten exponent. 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 percent on the average when p \equiv 1 (mod 16). Several numerical examples are given that show the speedup of the proposed methods.

[1] A.O.L. Atkin, Probabilistic Primality Testing, summary by F.Morain, Research Report 1779, INRIA, pp. 159-163, 1992.
[2] A.O.L. Atkin and F. Morain, “Elliptic Curves and Primality Proving,” Math. of Computation, vol. 61, pp. 29-68, 1993.
[3] D.V. Bailey, “Computation in Optimal Extension Fields,” master thesis, http://www.wpi.edu/Pubs/ETD/Available/etd-0428100-133037/ unrestrictedbailey.pdf , 2008.
[4] P.S.L.M. Barreto, H.Y. Kim, B. Lynn, and M. Scott, “Efficient Algorithms for Pairing-Based Cryptosystems,” Proc. 22nd Ann. Int'l Cryptology Conf. (CRYPTO '02), pp. 354-368, 2002.
[5] P.S.L.M. Barreto and J.F. Voloch, “Efficient Computation of Roots in Finite Fields,” J. Design, Codes and Cryptography, vol. 39, pp. 275-280, 2006.
[6] D. Boneh and M. Franklin, “Identity-Based Encryption from the Weil Pairing,” Proc. 21st Ann. Int'l Cryptology Conf. (CRYPTO '01), pp. 213-229, 2001.
[7] R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective. Springer, 2001.
[8] 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.
[9] Standard Specifications for Public Key Cryptography, IEEEStd2000-1363, 2000.
[10] N. Koblitz, “Elliptic Curve Cryptosystems,” Math. of Computation, vol. 48, pp. 203-209, 1987.
[11] 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, no. 1, pp. 1-5, 2006.
[12] C. Lee and J. Lee, “A Scalable Structure for a Multiplier and an Inversion Unit in $GF(2^{m})$ ,” ETRI J., vol. 25, no. 5, pp. 315-320, Oct. 2003.
[13] R. Lidl and H. Niederreiter, “Finite Field,” Encyclopedia of Math. and Its Applications, vol. 20, Cambridge Univ. Press, 1997.
[14] S. Lindhurst, “An Analysis of Shanks's Algorithm for Computing Square Roots in Finite Fields,” CRM Proc. and Lecture Notes, vol. 19, pp. 231-242, 1999.
[15] A. Menezes, P. van Oorschot, and S. Vanstone, Handbook of Applied Cryptography. CRC Press, 1997.
[16] V.S. Miller, “Use of Elliptic Curves in Cryptography,” Proc. Advances in Cryptology (CRYPTO '85), pp. 417-426, 1986.
[17] S. Müller, “On the Computation of Square Roots in Finite Fields,” J. Design, Codes and Cryptography, vol. 31, pp. 301-312, 2004.
[18] Y. Nogami, M. Obara, and Y. Morikawa, “A Method for Distinguishing the Two Candidate Elliptic Curves in the Complex Multiplication Method,” ETRI J., vol. 28, no. 6, pp.745-760, Dec. 2006.
[19] R. Schoof, “Counting Points on Elliptic Curves over Finite Fields,” J. Th'eorie des Nombres de Bordeaux, vol. 7, pp. 219-254, 1995.
[20] D. Shanks, “Five Number-Theoretic Algorithms,” Proc. Second Manitoba Conf. Numerical Math., pp. 51-70, 1972.
[21] A. Tonelli, “Bemerkung über die Auflösung Quadratischer Congruenzen,” Göttinger Nachrichten, pp. 344-346, 1891.

Index Terms:
Square roots, finite fields, efficient computation, cryptography.
Citation:
Dong-Guk Han, Dooho Choi, Howon Kim, "Improved Computation of Square Roots in Specific Finite Fields," IEEE Transactions on Computers, vol. 58, no. 2, pp. 188-196, Feb. 2009, doi:10.1109/TC.2008.201
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