Bounded Simultaneous Messages

TL;DR

This paper investigates bounded simultaneous messages protocols where unbounded Alice and Bob send polynomial-size messages to a bounded Carol to evaluate functions, exploring the limits especially for NP-complete problems.

Contribution

It systematically studies the BSM model with various functions and bounds, providing evidence against polynomial-size protocols for certain NP-related problems.

Findings

No polynomial-size BSM protocols for some NP-complete variants.

Initiates systematic study of BSM with different functions and bounds.

Highlights limitations in distributed computation models.

Abstract

We consider the following question of bounded simultaneous messages (BSM) protocols: Can computationally unbounded Alice and Bob evaluate a function $f(x,y)$ of their inputs by sending polynomial-size messages to a computationally bounded Carol? The special case where $f$ is the mod-2 inner-product function and Carol is bounded to AC$^0$ has been studied in previous works. The general question can be broadly motivated by applications in which distributed computation is more costly than local computation, including secure two-party computation. In this work, we initiate a more systematic study of the BSM model, with different functions $f$ and computational bounds on Carol. In particular, we give evidence against the existence of BSM protocols with polynomial-size Carol for naturally distributed variants of NP-complete languages.