On the multiplicative complexity of Boolean functions over the basis ∧,⊕,1

The multiplicative complexity $c_{\wedge}(f)$ of a Boolean function $f$ is the minimum number of AND gates in a circuit representing $f$ which employs only AND, XOR and NOT gates. A constructive upper bound, $c_{\wedge}(f) = 2^{\frac{n}{2} + 1}-n/2 -2$, for any Boolean function $f$ on $n$ variables ($n$ even) is given. A counting argument gives a lower bound of $c_{\wedge}(f) = 2^{\frac{n}{2}} - O(n)$. Thus we have shown a separation, by an exponential factor, between worst-case Boolean complexity (which is known to be $\Theta(2^n)$) and worst-case multiplicative complexity. A construction of circuits for symmetric Boolean functions on $n$ variables, requiring less than $n + \frac{3}{2} \sqrt{n}$ AND gates, is described.