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On Explicit Constructions of Extremely Depth Robust Graphs

Abstract

A directed acyclic graph G=(V,E)G=(V,E) is said to be (e,d)(e,d)-depth robust if for every subset SVS \subseteq V of Se|S| \leq e nodes the graph GSG-S still contains a directed path of length dd. If the graph is (e,d)(e,d)-depth-robust for any e,de,d such that e+d(1ϵ)Ve+d \leq (1-\epsilon)|V| then the graph is said to be ϵ\epsilon-extreme depth-robust. In the field of cryptography, (extremely) depth-robust graphs with low indegree have found numerous applications including the design of side-channel resistant Memory-Hard Functions, Proofs of Space and Replication, and in the design of Computationally Relaxed Locally Correctable Codes. In these applications, it is desirable to ensure the graphs are locally navigable, i.e., there is an efficient algorithm GetParents\mathsf{GetParents} running in time polylogV\mathrm{polylog} |V| which takes as input a node vVv \in V and returns the set of vv's parents. We give the first explicit construction of locally navigable ϵ\epsilon-extreme depth-robust graphs with indegree O(logV)O(\log |V|). Previous constructions of ϵ\epsilon-extreme depth-robust graphs either had indegree ω~(log2V)\tilde{\omega}(\log^2 |V|) or were not explicit.

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