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**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

A directed acyclic graph G = (V,E) is said to be (e,d)-depth robust if for every subset S ⊆ V of |S| ≤ 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 ≤ (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 GetParents running in time polylog|V| which takes as input a node v ∈ 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 ω̃(log² |V|) or were not explicit.

Jeremiah Blocki, Mike Cinkoske, Seunghoon Lee, and Jin Young Son. On Explicit Constructions of Extremely Depth Robust Graphs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 14:1-14:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{blocki_et_al:LIPIcs.STACS.2022.14, author = {Blocki, Jeremiah and Cinkoske, Mike and Lee, Seunghoon and Son, Jin Young}, title = {{On Explicit Constructions of Extremely Depth Robust Graphs}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {14:1--14:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.14}, URN = {urn:nbn:de:0030-drops-158241}, doi = {10.4230/LIPIcs.STACS.2022.14}, annote = {Keywords: Depth-Robust Graphs, Explicit Constructions, Data-Independent Memory Hard Functions, Proofs of Space and Replication} }

Document

**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

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 ⊂ V (resp. S ⊆ E) of at most |S| ≤ 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 ((n log log n)/log n, n/{(log n)^{1 + log log n}})-depth-robust graph with constant indegree, where previous constructions for e = (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 (k₁, k₂)-ST-Robust if we can remove any k₁ nodes and there exists a subgraph containing at least k₂ inputs and k₂ outputs such that each of the k₂ inputs is connected to all of the k₂ outputs. If the graph if (k₁,n-k₁)-ST-Robust for all k₁ ≤ 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 𝕄 of ST-robust graphs and an arbitrary (e, d)-edge-depth-robust graph G we construct a new constant-indegree graph Reduce(G, 𝕄) by replacing each node in G with an ST-robust graph from 𝕄. We also show that ST-robust graphs can be used to construct (tight) proofs-of-space and (asymptotically) improved wide-block labeling functions.

Jeremiah Blocki and Mike Cinkoske. A New Connection Between Node and Edge Depth Robust Graphs. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 64:1-64:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{blocki_et_al:LIPIcs.ITCS.2021.64, author = {Blocki, Jeremiah and Cinkoske, Mike}, title = {{A New Connection Between Node and Edge Depth Robust Graphs}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {64:1--64:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.64}, URN = {urn:nbn:de:0030-drops-136038}, doi = {10.4230/LIPIcs.ITCS.2021.64}, annote = {Keywords: Depth robust graphs, memory hard functions} }

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