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-rank, 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：假设两位玩家打算计算一个布尔函数$f(x,y)$，其中第一位玩家已知$x$，第二位玩家已知$y$，他们可以发送和接收用比特编码的消息，那么他们可以通过交换$(\log \rank (M_f))^c $比特来计算$f(x,y)$，其中$M_f$是由函数$f$确定的布尔矩阵。该问题广泛开放且备受关注，在过去35年间屡次被尝试攻克却未果。目前最佳上界仍呈指数级依赖猜想中的界。遗憾的是，我们无法证明该猜想，但我们提出了一种通信协议，使用$(\log \rank (M_f))^c $比特来计算与$f(x,y)$——在某种程度上——相关的量。这种相关性通过模合数的低秩、多元多项式表示来刻画。我们的这一结果可能有助于解决这个长期悬而未决的猜想。