On Explicit Constructions of Extremely Depth Robust Graphs

TL;DR

This paper presents the first explicit construction of extremely depth-robust graphs that are locally navigable and have low indegree, addressing key needs in cryptography and related fields.

Contribution

The paper introduces the first explicit, locally navigable, epsilon-extreme depth-robust graphs with indegree O(log |V|), improving over previous non-explicit or higher indegree constructions.

Findings

Constructed explicit graphs with low indegree and extreme depth-robustness.

Ensured graphs are locally navigable with polylogarithmic time algorithms.

Achieved graphs suitable for cryptographic applications like Memory-Hard Functions.

Abstract

A directed acyclic graph $G=(V,E)$ is said to be $(e,d)$-depth robust if for every subset $S \subseteq V$ of $|S| \leq e$ nodes the graph $G-S$ still contains a directed path of length $d$. If the graph is $(e,d)$-depth-robust for any $e,d$ such that $e+d \leq (1-\epsilon)|V|$ then the graph is said to be $\epsilon$-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 $\mathsf{GetParents}$ running in time $\mathrm{polylog} |V|$ which takes as input a node $v \in V$ and returns the set of $v$'s parents. We give the first explicit construction of locally navigable $\epsilon$-extreme depth-robust graphs with indegree $O(\log |V|)$. Previous constructions of $\epsilon$-extreme depth-robust graphs either had indegree $\tilde{\omega}(\log^2 |V|)$ or were not explicit.