On the algebraic thickness and non-normality of Boolean functions

Cryptographic Boolean functions must be complex to satisfy Shannon's principle of confusion. From the cryptographic viewpoint, the two main criteria in evaluating the complexity of Boolean functions on F/sub 2//sup n/ are the nonlinearity and the algebraic degree. Two other criteria have also been considered: the algebraic thickness and the non-normality. It is known that, asymptotically, almost all Boolean functions have high algebraic thicknesses and are deeply non-normal, and, as well, they have high algebraic degrees and high nonlinearities. We improve upon this result and, recalling a relationship between non-normality and nonlinearity, we prove a new result on symmetric functions, which implies, as a direct consequence, the known results on their nonlinearities (this gives some new insight on the reasons for their behavior).