The problem I consider is the following: given the $2^n$ coefficients of a Boolean polynomial $f : \{0, 1\}^n \rightarrow \{0, 1\}$, determine if $f$ is balanced namely if the truth table of $f$ contains exactly $2^{n-1}$ zeroes and $2^{n-1}$ ones. (in other terms: determine if a Boolean function is balanced given its algebraic normal form).
Now assume that two parties, Alice and Bob, each possess 50% of the input coefficients table (they each posses $2^{n-1}$ bits, but the subset of the coefficients that Alice has can be somewhat arbitrary).
If Alice simply sends her input to Bob, they can determine if the input is a balanced function with a communication of $2^{n-1}$ bits.
My question is: how to prove that they can't do anything really better than this? I'll be happy with a proof that at least $\alpha 2^n$ bits are necessary, for any constant $\alpha > 0$.
