×

Recursion orders for weights of Boolean cubic rotation symmetric functions. (English) Zbl 1384.94049

Summary: Rotation symmetric (RS) Boolean functions have been extensively studied in recent years because of their applications in cryptography. In cryptographic applications, it is usually important to know the weight of the functions, so much research has been done on the problem of determining such weights. Recently it was proved that for cubic RS functions in \(n\) variables generated by a single monomial, the weights of the functions as \(n\) increases satisfy a linear recursion. Furthermore, explicit methods were found for generating these recursions and the initial values needed to use the recursions. It is important to be able to compute the order of these recursions without needing to determine all of the coefficients. This paper gives a technique for doing that in many cases, based on a new notion of towers of RS Boolean functions.

MSC:

94D10 Boolean functions
94A60 Cryptography
05E05 Symmetric functions and generalizations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bileschi, M. L.; Cusick, T. W.; Padgett, D., Weights of Boolean cubic monomial rotation symmetric functions, Cryptogr. Commun., 4, 105-130 (2012) · Zbl 1282.94109
[2] Brown, A.; Cusick, T. W., Equivalence classes for cubic rotation symmetric functions, Cryptogr. Commun., 5, 85-118 (2013) · Zbl 1335.94112
[3] Brown, A.; Cusick, T. W., Recursive weights for some boolean functions, J. Math. Cryptol., 6, 105-135 (2012) · Zbl 1277.94016
[5] Cusick, T. W., Affine equivalence of cubic homogeneous rotation symmetric boolean functions, Inform. Sci., 181, 5067-5083 (2011) · Zbl 1272.94026
[6] Cusick, T. W., Finding Hamming weights without looking at truth tables, Cryptogr. Commun., 5, 7-18 (2013) · Zbl 1335.94113
[7] Cusick, T. W.; Brown, A., Affine equivalence for rotation symmetric Boolean functions with \(p^k\) variables, Finite Fields Appl., 18, 547-562 (2012) · Zbl 1275.94026
[8] Cusick, T. W.; Cheon, Y., Affine equivalence of quartic homogeneous rotation symmetric boolean functions, Inform. Sci., 259, 192-211 (2014) · Zbl 1384.94048
[9] Cusick, T. W.; Padgett, D., A recursive formula for weights of boolean rotation symmetric functions, Discrete Appl. Math., 160, 391-397 (2011) · Zbl 1258.06011
[10] Cusick, Thomas W.; Staˇnicaˇ, Pantelimon, Cryptographic Boolean Functions (2009), Academic Press: Academic Press San Diego · Zbl 1173.94002
[11] Everest, G.; van der Poorten, A.; Shparlinski, I.; Ward, T., Recurrence sequences, (Math. Surveys Monographs, vol. 104 (2003)), American Mathematical Society, Providence · Zbl 1033.11006
[12] Kavut, S., Results on rotation-symmetric S-boxes, Inform. Sci., 201, 93-113 (2012) · Zbl 1276.94017
[13] Kavut, S.; Maitra, S.; Yücel, M. D., Search for boolean functions with excellent profiles in the rotation symmetric class, IEEE Trans. Inform. Theory, 53, 1743-1751 (2007) · Zbl 1287.94130
[14] Kavut, S.; Yücel, M. D., (Generalized Rotation Symmetric and Dihedral Symmetric Boolean Functions-9 variable Boolean Functions with Nonlinearity vol. 242. Generalized Rotation Symmetric and Dihedral Symmetric Boolean Functions-9 variable Boolean Functions with Nonlinearity vol. 242, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC 2007, LNCS, vol. 485 (2007), Springer Verlag), 321-329 · Zbl 1195.94062
[15] Kavut, S.; Yücel, M. D., 9-variable boolean functions with nonlinearity 242 in the generalized rotation symmetric class, Inform. Comput., 208, 341-350 (2010) · Zbl 1183.94039
[16] Kim, H.; Park, S.-M.; Hahn, S. G., On the weight and nonlinearity of homogeneous rotation symmetric boolean functions of degree 2, Discrete Appl. Math., 157, 428-432 (2009) · Zbl 1157.94005
[17] Maximov, A., (Classes of plateaued rotation symmetric Boolean functions under transformation of Walsh spectra. Classes of plateaued rotation symmetric Boolean functions under transformation of Walsh spectra, Workshop on Coding and Cryptography WCC 2005, LNCS, vol. 3969 (2006), Springer-Verlag), 325-334
[18] Wang, B.; Zhang, X.; Chen, W., The hamming weight and nonlinearity of a type of rotation symmetric boolean function, Acta Math. Sinica (Chin. Ser.), 55, 613-626 (2012) · Zbl 1274.06053
[19] Zhang, X.; Guo, H.; Feng, R.; Li, Y., Proof of a conjecture about rotation symmetric functions, Discrete Math., 311, 1281-1289 (2011) · Zbl 1235.94054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.