Third-order nonlinearities of a subclass of Kasami functions

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.