Cubic Boolean functions with highest resiliency
We classify those cubic m-variable Boolean functions which are (m - 4)-resilient. We prove that there are four types of such functions, depending on the structure of the support of their Walsh spectra. We are able to determine, for each type, the Walsh spectrum and, then, the nonlinearity of the corresponding functions. We also give the dimension of their linear space. This dimension equals m - k where k = 3 for the first type, k = 4 for the second type, k = 5 for the third type, and 5 less than or equal to k less than or equal to 9 for the fourth type.
boolean function cubic function hamming weight reed-muller code resilient function stream cipher symmetric cryptography