Boosting uniformity in quasirandom groups: fast and simple

TL;DR

This paper presents new exponential lower bounds on communication complexity for group multiplication in the number-on-forehead model, introduces simplified proofs, and extends results to general quasirandom groups and non-abelian settings.

Contribution

It provides the first exponential improvements in lower bounds, simplifies proofs, and generalizes uniformity results to quasirandom and non-abelian groups.

Findings

Exponential lower bounds on communication complexity in the number-on-forehead model.

Simplified proofs that extend to other quasirandom groups.

Generalization of uniformity approximation results to non-abelian groups.

Abstract

We study the communication complexity of multiplying $k\times t$ elements from the group $H=\text{SL}(2,q)$ in the number-on-forehead model with $k$ parties. We prove a lower bound of $(t\log H)/c^{k}$. This is an exponential improvement over previous work, and matches the state-of-the-art in the area. Relatedly, we show that the convolution of $k^{c}$ independent copies of a 3-uniform distribution over $H^{m}$ is close to a $k$-uniform distribution. This is again an exponential improvement over previous work which needed $c^{k}$ copies. The proofs are remarkably simple; the results extend to other quasirandom groups. We also show that for any group $H$, any distribution over $H^{m}$ whose weight-$k$ Fourier coefficients are small is close to a $k$-uniform distribution. This generalizes previous work in the abelian setting, and the proof is simpler.