A New Connection Between Node and Edge Depth Robust Graphs
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 (nloglognlogn,nlogn(logn)loglogn)-depth-robust graph with constant indegree. 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 more »
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10200741
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ITCS 2021
3. Memory-hard functions (MHFs) are a key cryptographic primitive underlying the design of moderately expensive password hashing algorithms and egalitarian proofs of work. Over the past few years several increasingly stringent goals for an MHF have been proposed including the requirement that the MHF have high sequential space-time (ST) complexity, parallel space-time complexity, amortized area-time (aAT) complexity and sustained space complexity. Data-Independent Memory Hard Functions (iMHFs) are of special interest in the context of password hashing as they naturally resist side-channel attacks. iMHFs can be specified using a directed acyclic graph (DAG) $G$ with $N=2^n$ nodes and low indegree and themore »