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Third-order nonlinearities of a subclass of Kasami functions

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Abstract

The rth-order nonlinearity, where r ≥ 1, of an n-variable Boolean function f, denoted by nl r (f), is defined as the minimum Hamming distance of f from all n-variable Boolean functions of degrees at most r. In this paper we obtain a lower bound of the third-order nonlinearities of Kasami functions of the form \(Tr_{1}^{n}(\mu x^{57})\). It is demonstrated that for large values of n the lower bound of the third-order nonlinearities of the functions of this form is larger than the general lower bound obtained by Carlet (IEEE Trans Inf Theory 54(3):1262–1272, 2008) for Kasami functions. Further we show that our result along with the computational results obtained by Fourquet and Tavernier (Designs Codes Cryptogr 49:323–340, 2008) provide us an estimate of the nonlinearity profiles of these functions for n = 7, 8, 10.

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Acknowledgement

The first author thanks theUniversity Grants Commission of India for supporting her research. This research is a part of the DST-JST strategic India-Japan Cooperative Program.

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Correspondence to Sugata Gangopadhyay.

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Gode, R., Gangopadhyay, S. Third-order nonlinearities of a subclass of Kasami functions. Cryptogr. Commun. 2, 69–83 (2010). https://doi.org/10.1007/s12095-009-0017-z

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