Multiplicative Complexity of Vector Valued Boolean Functions

Magnus Gausdal Find; Joan Boyar

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arXiv · 1407.6169

Multiplicative Complexity of Vector Valued Boolean Functions

Magnus Gausdal Find·Joan Boyar

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

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BibTeX RIS

Magnus Gausdal Find, Joan Boyar. 2018-02-22. Multiplicative Complexity of Vector Valued Boolean Functions. https://arxiv.org/abs/1407.6169

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