Sub-linear Secure Broadcast and Applications

We present improved distributed broadcast and MST algorithms that are unconditionally secure against an eavesdropper controlling a fixed set of at most f edges in an n-node m-edge D-diameter graph. We strive for secure algorithms with sublinear round and subquadratic message complexities (in n) for any f. This is in contrast to the exponential or polynomial dependence on f in prior works. Our main results are: Secure broadcast algorithm, for sending an O(logn)-bit message, that runs in Õ(D+√n) rounds and Õ(n3/2) messages. This matches the state-of-the-art bounds for insecure broadcast by [Ghaffari and Kuhn, and Gmyr and Pandurangan, DISC 2018]. Our bounds also improve over the Õ(D+√f n)-round complexity and Õ(√f n· m) message complexity of secure broadcast by [Hitron, Parter and Yogev, DISC 2022]. Secure MST algorithm with sublinear round and subcubic message complexities that improve over the algorithm by [Hitron, Parter and Yogev, ITCS 2023] in the entire regime. In particular, when f=(n), we improve the round complexity from Õ(n3/2) to Õ(n2/3), and the message complexity from Õ(n3) to Õ(n7/3). Our algorithms are randomized and their correctness and (statistical) security hold with high probability. The algorithms are based on a combination of techniques: Karger's sampling, tree packing and sparse recovery sketches.