On the multiplicative complexity of some Boolean functions

In this paper, we study the multiplicative complexity of Boolean functions. The multiplicative complexity of a Boolean function f is the smallest number of &-gates in circuits in the basis {x & y, x aS center dot y, 1} such that each such circuit computes the function f. We consider Boolean functions which are represented in the form x (1), x (2)a <-x (n) aS center dot q(x (1), a <-, x (n) ), where the degree of the function q(x (1), a <-, x (n) ) is 2. We prove that the multiplicative complexity of each such function is equal to (n - 1). We also prove that the multiplicative complexity of Boolean functions which are represented in the form x (1) a <- x (n) aS center dot r(x (1), a <-, x (n) ), where r(x (1), a <-, x (n) ) is a multi-affine function, is, in some cases, equal to (n - 1).