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Partial spread and vectorial generalized bent functions. (English) Zbl 1408.94997
Summary: In this paper we generalize the partial spread class and completely describe it for generalized Boolean functions from $${\mathbb {F}}_2^n$$ to $${\mathbb {Z}}_{2^t}$$. Explicitly, we describe gbent functions from $${\mathbb {F}}_2^n$$ to $${\mathbb {Z}}_{2^t}$$, which can be seen as a gbent version of Dillon’s $$PS_{ap}$$ class. For the first time, we also introduce the concept of a vectorial gbent function from $${\mathbb {F}}_2^n$$ to $${\mathbb {Z}}_q^m$$, and determine the maximal value which $$m$$ can attain for the case $$q=2^t$$. Finally we point to a relation between vectorial gbent functions and relative difference sets.

##### MSC:
 94C10 Switching theory, application of Boolean algebra; Boolean functions (MSC2010) 06E30 Boolean functions 05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
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##### References:
 [1] Çeşmelioğlu, A; McGuire, G; Meidl, W, A construction of weakly and non-weakly regular bent functions, J. Combin. Theory Ser. A, 119, 420-429, (2012) · Zbl 1258.94034 [2] Çeşmelioğlu A., Meidl W., Pott A.: Vectorial bent functions and their duals (manuscript). · Zbl 1361.11079 [3] Dillon J.F.: Elementary Hadamard difference sets. Ph.D. dissertation, University of Maryland (1974). · Zbl 0346.05003 [4] Hodz̆ić, S; Pasalic, E, Generalized bent functions—some general construction methods and related necessary and sufficient conditions, Cryptogr. Commun., 7, 469-483, (2015) · Zbl 1343.94064 [5] Kantor W.: Bent functions generalizing Dillon’s partial spread functions. arXiv:1211.2600v1. · Zbl 1367.94410 [6] Kumar, PV; Scholtz, RA; Welch, LR, Generalized bent functions and their properties, J. Combin. Theory Ser. A, 40, 90-107, (1985) · Zbl 0585.94016 [7] Lisonek, P; Lu, YH, Bent functions on partial spreads, Des. Codes Cryptogr., 73, 209-216, (2014) · Zbl 1355.94104 [8] Martinsen T., Meidl W., Stănică P.: Generalized bent functions and their Gray images. In: Proceedings of WAIFI (Gent 2016). Lecture Notes in Computer Science (to appear). · Zbl 1409.11135 [9] Nyberg K.: Perfect nonlinear S-boxes. In: Davies D.W. (ed.) Advances in Cryptology, EUROCRYPT ’91 (Brighton, 1991). Lecture Notes in Computer Science, vol. 547, pp. 378-386. Springer, Berlin (1991) · Zbl 0766.94012 [10] Rothaus, OS, On “bent” functions, J. Combin. Theory Ser. A, 20, 300-305, (1976) · Zbl 0336.12012 [11] Schmidt, KU, Quaternary constant-amplitude codes for multicode CDMA, IEEE Trans. Inform. Theory, 55, 1824-1832, (2009) · Zbl 1367.94344 [12] Schmidt, KU, $${\mathbb{Z}}_4$$-valued quadratic forms and quaternary sequence families, IEEE Trans. Inform. Theory, 55, 5803-5810, (2009) · Zbl 1367.94410 [13] Stănică, P; Martinsen, T; Gangopadhyay, S; Singh, BK, Bent and generalized bent Boolean functions, Des. Codes Cryptogr., 69, 77-94, (2013) · Zbl 1322.94094 [14] Tan, Y; Pott, A; Feng, T, Strongly regular graphs associated with ternary bent functions, J. Combin. Theory Ser. A, 117, 668-682, (2010) · Zbl 1267.05300 [15] Tang C., Xiang C., Qi Y., Feng K.: Complete characterization of generalized bent and $$2^k$$-bent Boolean functions. https://eprint.iacr.org/2016/335. · Zbl 1370.94614 [16] Zhang, WG; Pasalic, E, Highly nonlinear balanced S-boxes with good differential properties, IEEE Trans. Inform. Theory, 60, 7970-7979, (2014) · Zbl 1359.94634
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