Multiplicative complexity of vector valued Boolean functions

We consider the multiplicative complexity of Boolean functions with multiple bits of output, studying how large a multiplicative complexity is necessary and sufficient to provide a desired nonlinearity. For so-called ΣΠΣ circuits, we show that there is a tight connection between error correcting codes and circuits computing functions with high nonlinearity. Combining this with known coding theory results, we show that functions with n inputs and n outputs with the highest possible nonlinearity must have at least 2.32n AND gates. 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. Additionally we provide a function which, for general circuits, has multiplicative complexity at least 2n − 3. Finally we study the multiplicative complexity of “almost all” functions. We show that every function with n bits of input and m bits of output can be computed using at most 2.5(1 + o(1)) √ m2n AND gates. 1. Definitions and Preliminaries Let F2 be the finite field of order 2 and F n 2 the n-dimensional vector space over F2. We denote by [n] the set {1, . . . , n}. An (n,m)-function is a mapping from F2 to F m 2 and we refer to these as the Boolean functions. When m > 1 we say that the function is vector valued. It is well known that every (n, 1)-function f can be written uniquely as a multilinear polynomial over F2 f(x1, . . . , xn) = ∑