Algebraic normal form ($f(z)={⨁}_{S\subseteq [d]}{c}_S{\prod }_{i\in S}{z}_i$)

The algebraic normal form is the unique multilinear polynomial representation of a Boolean function over the field with two elements, using exclusive or for addition and ordinary bit multiplication for products. The empty product is one. Existence and uniqueness follow by induction from $f(z^{\prime },z_d)=f(z^{\prime },0)\oplus z_d(f(z^{\prime },0)\oplus f(z^{\prime },1))$: the first polynomial supplies all coefficients not containing $z_d$, and the difference supplies exactly those that do. For split inputs $(x,y)$, each monomial factors into one locally computed product of Alice's bits and one of Bob's bits, yielding a separated Boolean decomposition.