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Issue No.02 - February (2009 vol.58)
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.
Square roots, finite fields, efficient computation, cryptography.
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, February 2009, doi:10.1109/TC.2008.201
[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, 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.
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