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的安全多方计算（MPC）已在两种主要模型中研究：(1) 部分网络预先固定，因此腐败可依赖于其结构发生；(2) 通信图中的边作为协议的一部分动态确定。尽管大量文献已成功绘制出固定图模型中支持安全计算的图结构的可行性与局限性（包括强烈的经典下界），但这些下界不适用于后一种动态图设置——该设置近期出现了令人振奋的新结果，但仍相对未被充分探索。在本工作中，我们开启了在动态图模型中对MPC进行类似的基础性研究。作为第一步，我们研究了图扩展性这一性质。所有现有协议（隐式或显式）生成的通信图均为扩展图，但尚不清楚这是否本质如此。我们的结果包含两类（针对恒定比例的腐败）：* 上界：我们展示了在其诱导通信图不是扩展图的情况下（计算性、信息论性、低局部性，甚至同时具有低局部性与适应性安全性），在广泛设置中（均假设某种形式的输入无关初始化）存在的安全协议。* 下界：在无初始化模型（plain model）下，对于适应性腐败，我们证明对于某些功能函数，不存在任何协议能针对所有对抗策略维持非扩展通信图。我们的下界仅依赖协议正确性（而非隐私性），并需要出人意料的精细论证。更一般地，我们提供了一个分析MPC协议演化通信图的形式化框架，为研究安全计算与更广泛图性质之间的关系奠定起点。