The nonhomomorphicity of boolean functions

Abstract

We introduce the notion of nonhomomorphicity as an alternative criterion that forecasts nonlinear characteristics of a Boolean function. Although both nonhomomorphicity and nonlinearity reflect a “difference” between a Boolean function and all the affine functions, they are measured from diffierent perspectives. We are interested in nonhomomorphicity due to several reasons that include (1) unlike other criteria, we have not only established tight lower and upper bounds on the nonhomomorphicity of a function, but also precisely identified the mean of nonhomomorphicity over all the Boolean functions on the same vector space, (2) the nonhomomorphicity of a function can be estimated efficiently, and in fact, we demonstrate a fast statistical method that works both on large and small dimensional vector spaces.