  The LogRank conjecture of Lov\'asz and Saks from 1988 is the most famous open
problem in the communication complexity theory. The statement is as follows:
Suppose that two players intend to compute a Boolean function $f(x,y)$ when $x$
is known for the first and $y$ for the second player, and they may send and
receive messages encoded with bits, then they can compute $f(x,y)$ with
exchanging $(\log \rank (M_f))^c $ bits, where $M_f$ is a Boolean matrix,
determined by function $f$. The problem is widely open and very popular, and it
has resisted numerous attacks in the last 35 years. The best upper bound is
still exponential in the bound of the conjecture. Unfortunately, we cannot
prove the conjecture, but we present a communication protocol with $(\log \rank
(M_f))^c $ bits, which computes a -- somewhat -- related quantity to $f(x,y)$.
The relation is characterized by a representation of low-degree, multi-linear
polynomials modulo composite numbers. This result of ours may help to settle
this long-time open conjecture.
