Secure Computation Meets Distributed Universal Optimality

We present a new algorithmic approach to distributed secure algorithms that is based on combining two independent lines of research: secure computation and distributed universal optimality. Our end result provides round-efficient distributed algorithms that protect the privacy of the graph vertices against a (possibly large) coalition of semi-honest adversaries.Secure Computation: The notion of perfect privacy dates back to Yao [FOCS '82], and has been extensively addressed by the Cryptographic community over the years. Most of the prior work considers the Multi-Party-Communication (MPC) model, in which the parties are fully connected. Considerably less is known on the (round) complexity of secure algorithms for general graphs, especially under the classical message passing models, such as the CONGEST model. For any biconnected D-diameter graph, a recent line of works [Parter and Yogev, SODA 19, ICALP 19, PODC 19] presented a simulation result that protects against a single semi-honest corruption, by paying an overhead of D · poly(Δ) CONGEST rounds, where Δ is the maximum degree. Due to an inherent structural barrier, the generalization of the current framework to handling f corruptions, provably leads to an overhead of O(Δ D)Θ(f) rounds. This can also be shown to be tight for the class of store-and-forward algorithms¹¹In which nodes can only propagate messages as atomic units, without the ability to mix multiple messages together.Secure Computation & Universal Optimality. We present an improved framework for secure computation which bypasses the current exponential in f barrier. For every graph G=(V, E) with vertex-connectivity Ω(f), our simulation provides a round overhead of poly(Δ) · O(S Q(G)).² The graph measure S Q(G) (Shortcut Quality) captures the universal optimal complexity of many network optimization tasks in the (non-secure) CONGEST model, as demonstrated in a recent breakthrough result of [Haeupler, Wajc and Zuzic, STOC 2021]. We are hopeful that the extended graph-theoretic machinery provided in this paper will find further applications in secure computation and beyond.²The notation O(·) hides 2O(√log n) factors.