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On the existence and non-existence of some classes of bent-negabent functions. (English) Zbl 1492.94251

Summary: In this paper we investigate different questions related to bent-negabent functions. We first take an expository look at the state-of-the-art research in this domain and point out some technical flaws in certain results and fix some of them. Further, we derive a necessary and sufficient condition for which the functions of the form \(\mathbf{x}\cdot \pi (\mathbf{y})\oplus h(\mathbf{y})\) [Maiorana-McFarland \((\mathcal{M})]\) is bent-negabent, and more generally, we study the non-existence of bent-negabent functions in the \(\mathcal{M}\) class. We also identify some functions that are bent-negabent. Next, we continue the recent work by B. Mandal et al. [Discrete Appl. Math. 236, 1–6 (2018; Zbl 1431.94217)] on rotation symmetric bent-negabent functions and show their non-existence in larger classes. For example, we prove that there is no rotation symmetric bent-negabent function in \(4p^k\) variables, where \(p\) is an odd prime. We present the non-existence of such functions in certain classes that are affine transformations of rotation symmetric functions. Keeping in mind the existing literature, we correct here some technical issues and errors found in other papers and provide some novel results.

MSC:

94D10 Boolean functions

Citations:

Zbl 1431.94217

Software:

GIFT
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Full Text: DOI

References:

[1] Banik, S., Pandey, S.K., Peyrin, T., Sasaki, Y., Sim, S., Siang, M., Todo, Y.: GIFT: a small present towards reaching the limit of lightweight encryption. In: CHES 2017. LNCS, vol. 10529, pp. 321-345 (2017) · Zbl 1450.94026
[2] Canteaut, A.; Charpin, P., Decomposing bent functions, IEEE Trans. Inf. Theory, 49, 8, 2004-2019 (2003) · Zbl 1184.94230 · doi:10.1109/TIT.2003.814476
[3] Carlet, C.; Crama, Y.; Hammer, P., Boolean functions for cryptography and error correcting codes, Boolean Methods and Models, 257-397 (2010), Cambridge: Cambridge University Press, Cambridge · Zbl 1209.94035
[4] Carlet, C., On the secondary constructions of resilient and bent functions, Coding Cryptogr. Comb., 23, 3-28 (2004) · Zbl 1062.94036
[5] Carlet, C.; Mesnager, S., Four decades of research on bent functions, Des. Codes Cryptogr., 78, 1, 5-50 (2016) · Zbl 1378.94028 · doi:10.1007/s10623-015-0145-8
[6] Cusick, TW; Stănică, P., Cryptographic Boolean Functions and Applications (2017), San Diego: Academic Press, San Diego · Zbl 1359.94001
[7] Dalai, DK; Maitra, S.; Sarkar, S., Results on rotation symmetric bent functions, Disc Math., 309, 8, 2398-2409 (2009) · Zbl 1171.94004 · doi:10.1016/j.disc.2008.05.017
[8] Dillon, J.F.: A survey of bent functions. NSA Tech. J. NSAL-S-203(92), 191-215 (1972) (Special Issue)
[9] Kavut, S.; Maitra, S.; Yücel, MD, Search for Boolean functions with excellent profiles in the rotation symmetric class, IEEE Trans. Inf. Theory, 53, 5, 1743-1751 (2007) · Zbl 1287.94130 · doi:10.1109/TIT.2007.894696
[10] McFarland, RL, A family of noncyclic difference sets, J. Comb. Theory Ser. A, 15, 1-10 (1973) · Zbl 0268.05011 · doi:10.1016/0097-3165(73)90031-9
[11] Mandal, B.; Singh, B.; Gangopadhyay, S.; Maitra, S.; Vetrivel, V., On non-existence of bent-negabent rotation symmetric Boolean functions, Discrete Appl. Math., 236, 1-6 (2018) · Zbl 1431.94217 · doi:10.1016/j.dam.2017.11.001
[12] Mesnager, S., Several new infinite families of bent functions and their duals, IEEE Trans. Inf. Theory, 60, 7, 4397-4407 (2014) · Zbl 1360.94480 · doi:10.1109/TIT.2014.2320974
[13] Mesnager, S.: Bent Functions—Fundamentals and Results, pp. 1-544. Springer, Bern (2016). ISBN 978-3-319-32593-4 · Zbl 1364.94008
[14] Parker, M.G.: The constabent properties of Goley-Devis-Jedwab sequences. In: International Symposium on Information Theory, Sorrento, Italy (2000). http://www.ii.uib.no/ matthew/. Accessed 9 June 2020
[15] Parker, M.G., Pott, A.: On Boolean functions which are bent and negabent. In: Golomb, S.W., Gong, G., Helleseth, T., Song, H.Y. (eds.) Sequences, Subsequences, and Consequences, SSC 2007 LNCS, vol. 4893, pp. 9-23 (2007) · Zbl 1154.94426
[16] Pieprzyk, J.; Qu, CX, Fast hashing and rotation-symmetric functions, J. Univ. Comput. Sci., 5, 1, 20-31 (1999)
[17] Rothaus, OS, On bent functions, J. Comb. Theory Ser. A, 20, 300-305 (1976) · Zbl 0336.12012 · doi:10.1016/0097-3165(76)90024-8
[18] Riera, C.; Parker, MG, Generalized bent criteria for Boolean functions, IEEE Trans. Inf. Theory, 52, 9, 4142-4159 (2006) · Zbl 1323.94137 · doi:10.1109/TIT.2006.880069
[19] Schmidt, K.-U., Parker, M.G., Pott, A.: Negabent functions in Maiorana-McFarland class. In: SETA, LNCS 2008, vol. 5203, pp. 390-402 (2008) · Zbl 1206.94090
[20] Sarkar, S.: Characterizing negabent Boolean functions over finite fields. In: Proceedings of SETA 2012, LNCS, vol. 7280, pp. 77-88 (2012) · Zbl 1290.94172
[21] Sarkar, S., Cusick, T.W.: Initial results on the rotation symmetric bent-negabent functions. In: 7th International Workshop on Signal Design and Applications in Communications (IWSDA), pp. 80-84 (2015)
[22] Stănică, P.; Gangopadhyay, S.; Chaturvedi, A.; Kar Gangopadhyay, A.; Maitra, S., Investigations on bent and negabent functions via the nega-Hadamard transform, IEEE Trans. Inf. Theory, 58, 6, 4064-4072 (2012) · Zbl 1365.94684 · doi:10.1109/TIT.2012.2186785
[23] Stănică, P.; Maitra, S., Rotation symmetric Boolean functions—count and cryptographic properties, Discrete Appl. Math., 156, 1567-1580 (2008) · Zbl 1142.94016 · doi:10.1016/j.dam.2007.04.029
[24] Stănică, P.; Mandal, B.; Maitra, S., The connection between quadratic bent-negabent functions and the Kerdock code, Appl. Algebra Eng. Commun. Comput., 30, 5, 387-401 (2019) · Zbl 1457.94252 · doi:10.1007/s00200-019-00380-4
[25] Su, W.; Pott, A.; Tang, X., Characterization of negabent functions and construction of bent-negabent functions with maximum algebraic degree, IEEE Trans. Inf. Theory, 59, 6, 3387-3395 (2013) · Zbl 1364.94806 · doi:10.1109/TIT.2013.2245938
[26] Xia, T.; Seberry, J.; Pieprzyk, J.; Charnes, C., Homogeneous bent functions of degree \(n\) in \(2n\) variables do not exist for \(n > 3\), Discrete Appl. Math., 142, 1-3, 127-132 (2004) · Zbl 1048.94016 · doi:10.1016/j.dam.2004.02.006
[27] Zhang, F.; Wei, Y.; Pasalic, E., Constructions of bent-negabent functions and their relation to the completed Maiorana-McFarland class, IEEE Trans. Inf. Theory, 61, 3, 1496-1506 (2015) · Zbl 1359.94939 · doi:10.1109/TIT.2015.2393879
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