We propose a linear algebraic method, rooted in the spectral properties of graphs, that can be used to prove lower bounds in communication complexity. Our proof technique effectively marries spectral bounds with information-theoretic inequalities. The key insight is the observation that, in specific settings, even when data sets $X$ and $Y$ are closely correlated and have high mutual information, the owner of $X$ cannot convey a reasonably short message that maintains substantial mutual information with $Y$. In essence, from the perspective of the owner of $Y$, any sufficiently brief message $m=m(X)$ would appear nearly indistinguishable from a random bit sequence. We employ this argument in several problems of communication complexity. Our main result concerns cryptographic protocols. We establish a lower bound for communication complexity of multi-party secret key agreement with unconditional, i.e., information-theoretic security. Specifically, for one-round protocols (simultaneous messages model) of secret key agreement with three participants we obtain an asymptotically tight lower bound. This bound implies optimality of the previously known omniscience communication protocol (this result applies to a non-interactive secret key agreement with three parties and input data sets with an arbitrary symmetric information profile). We consider communication problems in one-shot scenarios when the parties' inputs are not produced by any i.i.d. sources, and there are no ergodicity assumptions on the input data. In this setting, we found it natural to present our results using the framework of Kolmogorov complexity.

翻译：我们提出一种基于图谱性质的线性代数方法，可用于证明通信复杂度的下界。我们的证明技术有效融合了谱界与信息论不等式。关键洞察在于：在特定场景中，即使数据集$X$和$Y$高度相关且互信息量很大，当$X$数据持有者无法传递足够短的消息来维持与$Y$的显著互信息。本质上，从$Y$持有者视角看，任何足够短的编码消息$m=m(X)$都将近似无法与随机比特序列区分。我们将此论证应用于多个通信复杂度问题中。主要结果涉及密码协议：我们建立了具有无条件（即信息论安全）多方密钥协商通信复杂度的下界。特别地，针对三方密钥协商的单轮协议（同步消息模型），我们获得了渐进紧的下界。该下界证实了先前已知的全知通信协议的最优性（该结果适用于三方非交互式密钥协商，且输入数据集具有任意对称信息分布）。我们考虑单次场景下的通信问题，此时参与方的输入并非由独立同分布源生成，且不对输入数据作遍历性假设。在此设定下，我们自然采用柯尔莫戈洛夫复杂度框架来呈现结果。