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$f(x,y)$？特殊情况涉及$f$为模2内积函数且Carol被限制为AC$^0$，已在先前工作中得到研究。该一般性问题可广泛适用于分布式计算比本地计算成本更高的应用场景，包括安全双方计算。在本工作中，我们系统性地开展了对BSM模型的研究，涵盖不同函数$f$及Carol的计算能力约束。特别地，我们给出了针对NP完全语言的自然分布式变体不存在具有多项式大小Carol的BSM协议的证据。