Multiplicative Complexity of Vector Valued Boolean Functions

TL;DR

This paper investigates the multiplicative complexity of vector-valued Boolean functions, establishing bounds related to nonlinearity, connecting circuit complexity with coding theory, and analyzing typical function complexities.

Contribution

It introduces tight bounds on the multiplicative complexity for high nonlinearity functions and links circuit complexity to error-correcting codes, providing new theoretical insights.

Findings

Functions with maximum nonlinearity require at least 2.32n AND gates.

A bilinear circuit can compute a high nonlinearity function with complexity equal to the shortest code length.

Almost all functions can be computed with approximately 2.5√(m2^n) AND gates.

Abstract

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 $\Sigma\Pi\Sigma$ 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 $2.5(1+o(1))\sqrt{m2^n}$ AND gates.