Best Affine and Quadratic Approximations of Particular Classes of Boolean Functions

In this paper, we consider the problem of computing best low-order approximations of Boolean functions; we focus on the best quadratic approximations of a subclass of cubic functions with arbitrary number of variables and we provide formulas for their efficient calculation. Our methodology is developed upon properties of the best affine approximations of quadratic functions, for which formulas for their direct computation (not by means of the Walsh-Hadamard transform) are given. We determine the cubic functions in the above subclass that achieve the maximum second-order nonlinearity, thus yielding a lower bound for the covering radius of the second order Reed-Muller code ssr RM(2, n) in ssr RM(3, n). Simple extensions of these results to some special cases of higher degree functions, are seen to hold. Furthermore, a preliminary analysis of well-known constructions for bent functions, in terms of their second-order nonlinearity, is performed that indicates potential weaknesses if construction parameters are not properly chosen.