The Relationship between Multiplicative Complexity and Nonlinearity

We consider the relationship between nonlinearity and multiplicative complexity for Boolean functions with multiple outputs, studying how large a multiplicative complexity is necessary and sufficient to provide a desired nonlinearity. For quadratic circuits, we show that there is a tight connection between error correcting codes and circuits computing functions with high nonlinearity. Using known coding theory results, the lower bound proven here, for quadratic circuits for functions with n inputs and n outputs and high nonlinearity, shows that at least 2.32n AND gates are necessary. We further show that one cannot prove stronger lower bounds by only appealing to the nonlinearity of a function; we show a bilinear circuit computing a function with almost optimal nonlinearity with the number of AND gates being exactly the length of such a shortest code. For general circuits, we exhibit a concrete function with multiplicative complexity at least 2n − 3.