The multiplicative complexity of quadratic Boolean functions

Let the multiplicative complexity L(f) of a boolean function f be the minimal number of ∧-gates that are sufficient to evaluate f by circuits over the basis ∧,⊕,1. We give a polynomial time algorithm which for quadratic boolean forms f=⊕i≠jaijxixj determines L(f) from the coefficients aij. Two quadratic forms f,g have the same complexity L(f) = L(g) iff they are isomorphic by a linear isomorphism. We also determine the multiplicative complexity of pairs of quadratic boolean forms. We give a geometric interpretation to the complexity L(f1,f2) of pairs of quadratic forms.