Upper and Lower Bounds on Maximum Nonlinearity of n -input m -output Boolean Function

Abstract

The paper presents lower and upper bounds on the maximum nonlinearity for an n-input m-output Boolean function. We show a systematic construction method for a highly nonlinear Boolean function based on binary linear codes which contain the first order Reed-Muller code as a subcode. We also present a method to prove the nonexistence of some nonlinear Boolean functions by using nonexistence results on binary linear codes. Such construction and nonexistence results can be regarded as lower and upper bounds on the maximum nonlinearity. For some n and m, these bounds are tighter than the conventional bounds. The techniques employed here indicate a strong connection between binary linear codes and nonlinear n-input m-output Boolean functions.