1 research outputs found
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
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 inputs and
outputs with the highest possible nonlinearity must have at least
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 .
Finally we study the multiplicative complexity of "almost all" functions. We
show that every function with bits of input and bits of output can be
computed using at most AND gates.Comment: Extended version of the paper "The Relationship Between
Multiplicative Complexity and Nonlinearity", MFCS201