Multiplicative Complexity of Vector Value Boolean Functions

Published: September 21, 2015


Magnus G. Find, Joan Boyar


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.
Citation: arXiv e-Print Archive
Pub Type: Websites

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Nonlinearity, multiplicative complexity, circuit
Created September 21, 2015, Updated October 02, 2017