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Counting all bent functions in dimension eight 99270589265934370305785861242880. (English) Zbl 1215.94059
Summary: Based on the classification of the homogeneous Boolean functions of degree 4 in 8 variables, we present the strategy that we used to count the number of all bent functions in dimension 8. There are $$99270589265934370305785861242880 \approx 2^{106}$$ such functions in total. Furthermore, we show that most of the bent functions in dimension 8 are nonequivalent to Maiorana-McFarland and partial spread functions.

MSC:
 94D10 Boolean functions 06E30 Boolean functions 65T50 Numerical methods for discrete and fast Fourier transforms
Keywords:
Boolean functions; bent functions
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References:
 [1] Brier E., Langevin P.: Classification of the cubic forms of nine variables. In: PIEEE Information Theory Workshop La Sorbonne, Paris, France (2003). [2] Carlet C., Klapper A.: Upper bounds on the numbers of resilient functions and of bent functions. In: Werkgemeeschap voor Informatie- en Communicatietheorie, pp. 307–314 (2002). [3] Dillon J.F.: Elementary Hadamard Difference sets. PhD thesis, University of Maryland (1974). · Zbl 0346.05003 [4] Hou X.: GL(m, 2) acting on R(r, m)/R(r 1, m). Discret. Math. 149(1–3), 99–122 (1996) · Zbl 0852.94020 [5] Hou X., Langevin P.: Results on bent functions. J. Comb. Theory A 80(2), 232–246 (1997) · Zbl 0896.05011 [6] Hou X.: Cubic bent functions. Discret. Math. 189(1–3), 149–161 (1998) · Zbl 0949.94003 [7] Langevin P., Leander G.: Classification of the quartic forms of eight variables. In: Boolean Functions in Cryptology and Information Security, Svenigorod, Russia September (2007). · Zbl 1202.06009 [8] Langevin P., Leander G.: Classification of the quartic forms of eight variables. In: The Output of the Numerical Experiment, July 2007. http://www.univ-tln.fr/langevin/projects/quartics.html · Zbl 1202.06009 [9] Langevin P., Rabizzoni P., Véron P., Zanotti J.-P.: On the number of bent functions with 8 variables. In: Second International Workshop on Boolean Functions: Cryptography and Applications, pp. 125–135 (2006). [10] Maiorana J.A.: A classification of the cosets of the Reed-Muller code r(1, 6). Math. Comput. 57(195), 403–414 (1991) · Zbl 0724.94016 [11] McKay B.D.: Knight’s tours of an 8 {$$\times$$} 8 chessboard. Technical Report. TR-CS-97-03. Department of Computer Science, Australian National University (1997). [12] Rothaus O.S.: On ”bent” functions. J. Comb. Theory A 20(3), 300–305 (1976) · Zbl 0336.12012 [13] Sugita T., Kasami T., Fujiwara T.: Weight distributions of the third and fifth order Reed-Muller codes of length 512. Nara Institute of Science and Technology Report, February (1996). · Zbl 0862.94018 [14] Xiang-Dong H., Philippe L.: Counting partial spread bent functions. preprint (2009).
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