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

Subhamoy Maitra, Enes Pasalic
2006 Discrete Applied Mathematics  
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 2 n−1 − 2 n/2 + 2 n/2−2 . Here, we show that given any m, one can construct n-variable, m-resilient functions with nonlinearity 2 n−1 − 11 · 2 n/2−4 for all n 8m + 6 which is strictly greater than 2 n−1 − 2 n/2 + 2 n/2−2 . We
more » ... lso demonstrate that in some specific cases one may get such nonlinearity even for some values of n, where n < 8m+6. Further, we show that for sufficiently large n, it is possible to get such functions with nonlinearity reaching almost 2 n−1 − 2 n/2 + 4 3 2 n/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 2 n−3 .
doi:10.1016/j.dam.2005.03.014 fatcat:5aijo4u4qje2rkbcv24h3uczu4