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.

翻译：Lovász和Saks于1988年提出的LogRank猜想是通信复杂度理论中最著名的未解决问题。该猜想表述如下：假设两名玩家意图计算布尔函数$f(x,y)$，其中第一位玩家已知$x$、第二位玩家已知$y$，且双方可通过编码比特的消息进行收发，则存在$(\log \rank (M_f))^c$比特的交换使得双方能够计算$f(x,y)$，其中$M_f$是由函数$f$确定的布尔矩阵。该问题长期未解且广受关注，过去35年间虽经多次攻关仍未突破。目前最佳上界仍呈现猜想的指数级复杂度。遗憾的是，我们未能直接证明该猜想，但提出了一种通信协议，该协议使用$(\log \rank (M_f))^c$比特计算与$f(x,y)$存在——某种程度——关联的参量。这种关联通过模合数的低次多重线性多项式表示加以刻画。我们的这一结果或有助于解决这个长期悬而未决的猜想。