Secure multiparty computation (MPC) on incomplete communication networks has been studied within two primary models: (1) Where a partial network is fixed a priori, and thus corruptions can occur dependent on its structure, and (2) Where edges in the communication graph are determined dynamically as part of the protocol. Whereas a rich literature has succeeded in mapping out the feasibility and limitations of graph structures supporting secure computation in the fixed-graph model (including strong classical lower bounds), these bounds do not apply in the latter dynamic-graph setting, which has recently seen exciting new results, but remains relatively unexplored. In this work, we initiate a similar foundational study of MPC within the dynamic-graph model. As a first step, we investigate the property of graph expansion. All existing protocols (implicitly or explicitly) yield communication graphs which are expanders, but it is not clear whether this is inherent. Our results consist of two types (for constant fraction of corruptions): * Upper bounds: We demonstrate secure protocols whose induced communication graphs are not expander graphs, within a wide range of settings (computational, information theoretic, with low locality, even with low locality and adaptive security), each assuming some form of input-independent setup. * Lower bounds: In the plain model (no setup) with adaptive corruptions, we demonstrate that for certain functionalities, no protocol can maintain a non-expanding communication graph against all adversarial strategies. Our lower bound relies only on protocol correctness (not privacy), and requires a surprisingly delicate argument. More generally, we provide a formal framework for analyzing the evolving communication graph of MPC protocols, giving a starting point for studying the relation between secure computation and further, more general graph properties.

翻译：在不完全通信网络上进行安全多方计算（MPC）的研究主要基于两种模型：（1）部分网络结构预先固定，因此腐败行为可能依赖于该结构发生；（2）通信图中的边作为协议的一部分动态确定。尽管已有丰富的研究成功描述了固定图模型下支持安全计算的图结构的可行性与局限性（包括经典的下界结论），但这些下界不适用于后一种动态图场景——该场景近年来虽涌现出令人振奋的新成果，但仍相对未经探索。在本工作中，我们发起对动态图模型下MPC的类似基础性研究。作为第一步，我们探究图扩展性这一性质。所有现有协议（显式或隐式）产生的通信图均为扩展图，但尚不清楚这是否为本质特性。我们的研究结果包含两类（针对恒定腐败比例）：* 上界：在多种设置下（计算性、信息论安全、低局部性，甚至兼具低局部性与自适应安全性），我们展示了其诱导通信图并非扩展图的安全协议，每类协议均假设存在某种形式的输入无关预设结构。* 下界：在无预设结构的普通模型（plain model）中，针对自适应腐败，我们证明对于某些功能函数，任何协议都无法在所有对抗策略下维持非扩展通信图。此下界仅依赖协议的正确性（而非隐私性），且需要极为精巧的论证。更一般地，我们为分析MPC协议演化中的通信图提供了形式化框架，为研究安全计算与更广泛图性质之间的关系奠定基础。