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.

翻译：我们考虑以下有界同步消息（BSM）协议问题：计算能力无界的Alice和Bob能否通过向计算能力有界的Carol发送多项式大小的消息，来评估其输入上的函数$f(x,y)$？其中$f$为模2内积函数且Carol被限制为AC$^0$的特殊情况已在先前工作中得到研究。该一般性问题可广泛应用于分布式计算比本地计算成本更高的场景，包括安全两方计算。在本工作中，我们发起对BSM模型的更系统性研究，探讨不同的函数$f$以及Carol的计算能力限制。特别地，我们给出了NP完全语言自然分布变体不存在多项式规模Carol的BSM协议的证据。