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Ghosh, Kamara and Tamassia (ASIA CCS 2021) presented a Graph Encryption Scheme supporting shortest path queries. We show how to perform a query recovery attack against this GKT scheme when the adversary is given the original graph together with the leakage of certain subsets of queries. Our attack falls within the security model used by Ghosh et al., and is the first targeting schemes supporting shortest path queries. Our attack uses classical graph algorithms to compute the canonical names of the single-destination shortest path spanning trees of the underlying graph and uses these canonical names to pre-compute the set of candidate queries that match each response. Then, when all shortest path queries to a single node have been observed, the canonical names for the corresponding query tree are computed and the responses are matched to the candidate queries from the offline phase. The output is guaranteed to contain the correct query. For a graph on n vertices, our attack runs in time O(n 3 ) and matches the time complexity of the GKT scheme's setup. We evaluate the attack's performance using the real world datasets used in the original paper and show that as many as 21.9% of the queries can be uniquely recovered and as many as 50% of the queries result in sets of only three candidates.

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