A Maiorana–McFarland type construction for resilient Boolean functions on n variables (n even) with nonlinearity >2n-1-2n/2+2n/2-2>2n-1-2n/2+2n/2-2

In this paper, we present a construction method of m-resilient Boolean functions with very high nonlinearity for low values of m. The construction only considers functions in even number of variables n  . So far the maximum nonlinearity attainable by resilient functions was 2n-1-2n/2+2n/2-22n-1-2n/2+2n/2-2. Here, we show that given any m, one can construct n-variable, m  -resilient functions with nonlinearity 2n-1-11·2n/2-42n-1-11·2n/2-4 for all n⩾8m+6n⩾8m+6 which is strictly greater than 2n-1-2n/2+2n/2-22n-1-2n/2+2n/2-2. We also demonstrate that in some specific cases one may get such nonlinearity even for some values of n  , where n<8m+6n<8m+6. Further, we show that for sufficiently large n  , it is possible to get such functions with nonlinearity reaching almost 2n-1-2n/2+432n/2-2. This is the upper bound on nonlinearity when one uses our basic construction recursively. Lastly, we discuss the autocorrelation property of the functions and show that the maximum absolute value in the autocorrelation spectra is ⩽2n-3⩽2n-3.