Mike Cinkoske

Jeremiah Blocki, Mike Cinkoske, Seunghoon Lee et al.

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 $e+d \leq (1-ε)|V|$ then the graph is said to be $ε$-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 $ε$-extreme depth-robust graphs with indegree $O(\log |V|)$. Previous constructions of $ε$-extreme depth-robust graphs either had indegree $\tildeω(\log^2 |V|)$ or were not explicit.

Jeremiah Blocki, Mike Cinkoske

Given a directed acyclic graph (DAG) $G = (V,E)$, we say that $G$ is $(e,d)$-depth-robust (resp. $(e,d)$-edge-depth-robust) if for any set $S \subset V$ (resp. $S \subseteq E$) of at most $|S| \leq e$ nodes (resp. edges) the graph $G-S$ contains a directed path of length $d$. While edge-depth-robust graphs are potentially easier to construct many applications in cryptography require node depth-robust graphs with small indegree. We create a graph reduction that transforms an $(e, d)$-edge-depth-robust graph with $m$ edges into a $(e/2,d)$-depth-robust graph with $O(m)$ nodes and constant indegree. One immediate consequence of this result is the first construction of a provably $(\frac{n \log \log n}{\log n}, \frac{n}{(\log n)^{1 + \log \log n}})$-depth-robust graph with constant indegree, where previous constructions for $e =\frac{n \log \log n}{\log n}$ had $d = O(n^{1-ε})$. Our reduction crucially relies on ST-Robust graphs, a new graph property we introduce which may be of independent interest. We say that a directed, acyclic graph with $n$ inputs and $n$ outputs is $(k_1, k_2)$-ST-Robust if we can remove any $k_1$ nodes and there exists a subgraph containing at least $k_2$ inputs and $k_2$ outputs such that each of the $k_2$ inputs is connected to all of the $k_2$ outputs. If the graph if $(k_1,n-k_1)$-ST-Robust for all $k_1 \leq n$ we say that the graph is maximally ST-robust. We show how to construct maximally ST-robust graphs with constant indegree and $O(n)$ nodes. Given a family $ \mathbb{M}$ of ST-robust graphs and an arbitrary $(e, d)$-edge-depth-robust graph $G$ we construct a new constant-indegree graph $ \mathrm{Reduce}(G, \mathbb{M})$ by replacing each node in $G$ with an ST-robust graph from $ \mathbb{M}$. We also show that ST-robust graphs can be used to construct (tight) proofs-of-space and (asymptotically) improved wide-block labeling functions.