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Low Complexity Normal Elements over Finite Fields of Characteristic Two
July 2008 (vol. 57 no. 7)
pp. 990-1001
In this paper we extend previously known results on the complexities of normal elements. Using algorithms that exhaustively test field elements, we are able to provide the distribution of the complexity of normal elements for binary fields with degree extensions up to 39. We also provide current results on the smallest known complexity for the remaining degree extensions up to 330 by using a combination of constructive theorems and known exact values. We give an algorithm to exhaustively search field elements by using Gray codes that allows us to reuse previous computations, and compare this with the traditional method. We describe and analyze these algorithms and show both experimentally and asymptotically that the Gray code optimization gives substantial savings. The total computation of the distribution of the complexity of normal elements for degrees up to 39 in our experiments allows us to draw several conjectures. In particular, our data provides remarkable evidence for the conjecture that the complexity of normal elements follows a normal distribution. Finally, we propose that there is no linear bound on the minimum complexity with respect to the degree of the extension.

[1] D.W. Ash, I.F. Blake, and S.A. Vanstone, “Low Complexity Normal Bases,” Discrete Applied Math., vol. 25, pp. 191-210, 1989.
[2] T. Beth, W. Geiselmann, and F. Meyer, “Finding (Good) Normal Bases in Finite Fields,” Proc. Int'l Conf. Symbolic and Algebraic Computation, pp. 173-178, 1991.
[3] I.F. Blake, S. Gao, and R.C. Mullin, “Explicit Factorization of $x^{2^{k}} +1$ over ${\hbox{\rlap{I}\kern 2.0pt{\hbox{F}}}}_{p}$ with Prime $p \equiv 3 \bmod 4$ ,” Applicable Algebra in Eng., Comm., and Computing, vol. 4, pp. 89-94, 1993.
[4] M. Christopoulou, T. Garefalakis, D. Panario, and D. Thomson, “The Trace of an Optimal Normal Element and Low Complexity Normal Bases,” Designs, Codes and Cryptography, to appear, 2008.
[5] “Handbook of Elliptic and Hyperelliptic Curve Cryptography,” Discrete Math. and Its Applications, H. Cohen, G. Frey, R.M. Avanzi, C.Douche, T. Lange, K. Nguyen, and F. Vercauteren, eds., Chapman and Hall/CRC, 2006.
[6] G.S. Frandsen, “On the Density of Normal Bases in Finite Fields,” Finite Fields and Their Applications, vol. 6, pp. 23-38, 2000.
[7] S. Gao, J. von zur Gathen, D. Panario, and V. Shoup, “Algorithms for Exponentiation in Finite Fields,” J. Symbolic Computation, vol. 29, pp. 879-889, 2000.
[8] S. Gao and H.W. Lenstra, “Optimal Normal Bases,” Designs, Codes, and Cryptography, vol. 2, pp. 315-323, 1992.
[9] S. Gao and D. Panario, “Density of Normal Elements,” Finite Fields and Their Applications, vol. 3, pp. 141-150, 1997.
[10] J. von zur Gathen and J. Gerhard, Modern Computer Algebra, second ed. Cambridge Univ. Press, 2003.
[11] J. von zur Gathen and J. Gerhard, “Polynomial Factorization over ${\hbox{\rlap{I}\kern 2.0pt{\hbox{F}}}}_{2}$ ,” Math. of Computation, vol. 71, pp. 1677-1698, 2002.
[12] J. von zur Gathen and M. Giesbrecht, “Constructing Normal Bases in Finite Fields,” J. Symbolic Computation, vol. 10, pp. 547-570, 1990.
[13] W. Geiselmann, “Algebraische Algorithmenentwicklung am Beispiel der Arithmetik in endlichen Körpern,” dissertation, Universität Karlsruhe, Germany, 1992.
[14] K. Hensel, “Über die Darstellung der Zahlen eines Gattungsbereiches für einen beliebigen Primdivisor,” J. für die reine und angewandte Mathematik, vol. 103, pp. 230-237, 1888.
[15] D. Jungnickel, Finite Fields: Structure and Arithmetics. B.I. Wissenschaftsverlag, 1993.
[16] R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their Applications, second ed. Cambridge Univ. Press, 1994.
[17] F. Meyer, “Normalbasismultiplikation in endlichen Körpern; Diplomarbeit,” Univ. of Karlsruhe, Germany, 1990.
[18] R.C. Mullin, I.M. Onyszchuk, S.A. Vanstone, and R.M. Wilson, “Optimal Normal Bases in $GF(p^{n})$ ,” Discrete Applied Math., vol. 22, pp. 149-161, 1988/1989.
[19] M. Mendez and A. Pala, “Type I Error Rate and Power of Three Normality Tests,” Pakistan J. Information and Technology, vol. 2, pp.135-139, 2003.
[20] P. Ning and Y. Yin, “Efficient Software Implementation for Finite Field Multiplication in Normal Basis,” Lecture Notes in Computer Science, vol. 2229, pp. 177-188, 2001.
[21] C. Savage, “A Survey of Combinatorial Gray Codes,” SIAM Rev., vol. 39, pp. 605-629, 1997.
[22] V. Shoup, Number Theory Library (NTL) Version 5.4, http:/www.shoup.net, 2006.
[23] D. Stinson, “Some Observations on Parallel Algorithms for Fast Exponentiation in ${\hbox{\rlap{I}\kern 2.0pt{\hbox{F}}}}_{2^{n}}$ ,” SIAM J. Computing, vol. 19, pp. 711-717, 1990.
[24] C.C. Wang, “An Algorithm to Design Finite Field Multipliers Using a Self-Dual Normal Basis,” IEEE Trans. Computers, vol. 38, pp. 1457-1460, 1989.
[25] Z.-X. Wan and K. Zhou, “On the Complexity of the Dual Basis of a Type I Optimal Normal Basis,” Finite Fields and Their Applications, vol. 13, pp. 411-417, 2007.
[26] B. Young and D. Panario, “Low Complexity Normal Bases in ${\hbox{\rlap{I}\kern 2.0pt{\hbox{F}}}}_{2^{n}}$ ,” Finite Fields and Their Applications, vol. 10, pp. 53-64, 2004.

Index Terms:
Computations in finite fields, Computations on polynomials
Citation:
Ariane M. Masuda, Lucia Moura, Daniel Panario, David Thomson, "Low Complexity Normal Elements over Finite Fields of Characteristic Two," IEEE Transactions on Computers, vol. 57, no. 7, pp. 990-1001, July 2008, doi:10.1109/TC.2007.70845
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