Paper 2026/1220

Explicit Transformations from Edge-Depth-Robust to Node-Depth-Robust Graphs with Improved Concrete Efficiency

Abstract

Depth-robust directed acyclic graphs (DAGs) are an important combinatorial primitive in cryptography, with applications to memory-hard functions, proofs of space, and proofs of sequential work. These applications require node depth-robustness, yet constructing sparse graphs with strong concrete guarantees remains a longstanding challenge. By contrast, edge depth-robust graphs are easier to construct explicitly and achieve better parameters, motivating the problem of efficiently transforming edge-depth-robust graphs into node-depth-robust graphs. We give a new and substantially more efficient transformation from edge-depth-robust graphs to node-depth-robust graphs. We give a new and substantially more efficient transformation from edge-depth-robust graphs to node-depth-robust graphs. Prior work of Blocki and Cinkoske (ITCS 2021) gave a general transformation by replacing each node with an ST-robust graph, resulting in extraordinarily large overhead and relying on non-explicit components. In contrast, our transformation replaces each node with a single superconcentrator rather than an ST-robust graph, yielding an explicit construction whenever the input graph is explicit, with dramatically improved concrete efficiency. Formally, given an $(e,d)$-edge-depth-robust graph $G$ with $N$ nodes and $m$ edges, our transformation generates a graph $G'=Transform(G_N)$ with $N' = O(m)$ nodes, constant indegree, and $(e/3,\, 2d-1)$-node depth-robustness. We also show that the transformation preserves fractional depth-robustness. Moreover, when $m=\omega(N)$, we show that our transformation can amplify depth --- overcoming a key limitation of prior work. In particular, for any parameter $d'$, we can obtain a constant-indegree graph $G'$ with $N' = O(m+d'N)$ nodes and $(e/3,\, (d-e)d')$-node depth-robustness, yielding asymptotically tighter results in the important setting where $m = \omega(N)$ and $e \geq d$ by setting $d' \sim m/N$. As an application, We provide a novel analysis of Schnitger's edge-depth-robust graph construction (FOCS 1983). We show that the graph $G_n$, which has $N=2^{n+1}-1$ nodes and $m \leq n2^{n}$ edges, is $(e,2d-1)$-edge-depth robust for all $e + d \leq 2^n$. Applying our transformation yields an explicit DAG on $N'$ nodes with maximum indegree $2$ and depth-robustness parameters $e = \Omega(N'/\log N')$ and $d = \Omega(N')$, matching the best known asymptotic trade-offs—even compared with non-explicit constructions—while improving the best previously known concrete ed-product lower bound by a multiplicative factor of 6.6.