Simultaneous Multiparty Communication Complexity of Composed Functions

In the Number On the Forehead (NOF) multiparty communication model, $k$ players want to evaluate a function $F : X1 \times\cdots\times Xk\rightarrow Y$ on some input $(x1,\dots,xk)$ by broadcasting bits according to a predetermined protocol. The input is distributed in such a way that each player $i$ sees all of it except $xi$. In the simultaneous setting, the players cannot speak to each other but instead send information to a referee. The referee does not know the players' input, and cannot give any information back. At the end, the referee must be able to recover $F(x1,\dots,xk)$ from what she obtained. A central open question, called the $\log n$ barrier, is to find a function which is hard to compute for $polylog(n)$ or more players (where the $xi$'s have size $poly(n)$) in the simultaneous NOF model. This has important applications in circuit complexity, as it could help to separate $ACC^0$ from other complexity classes. One of the candidates belongs to the family of composed functions. The input to these functions is represented by a $k\times (t\cdot n)$ boolean matrix $M$, whose row $i$ is the input $xi$ and $t$ is a block-width parameter. A symmetric composed function acting on $M$ is specified by two symmetric $n$- and $kt$-variate functions $f$ and $g$, that output $f\circ g(M)=f(g(B1),\dots,g(Bn))$ where $Bj$ is the $j$-th block of width $t$ of $M$. As the majority function $MAJ$ is conjectured to be outside of $ACC^0$, Babai et. al. suggested to study $MAJ\circ MAJt$, with $t$ large enough. So far, it was only known that $t=1$ is not enough for $MAJ\circ MAJt$ to break the $\log n$ barrier in the simultaneous deterministic NOF model. In this paper, we extend this result to any constant block-width $t>1$, by giving a protocol of cost $2^{{O(2t)}\log{2{t+1}}(n)$} for any symmetric composed function when there are $2^{{\Omega(2t)}\log} n$ players.