Spectral Domain Analysis of Correlation Immune and Resilient Boolean Functions

We use a general property of Fourier transform to obtain direct proofs of recent divisibility results on the Walsh transform of correlation immune and resilient functions. Improved upper bounds on the nonlinearity of these functions are obtained from the divisibility results. We deduce further information on correlation immune and resilient functions. In particular, we obtain a necessary condition on the algebraic normal form of correlation immune functions attaining the maximum possible nonlinearity.