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Multi-step processing is commonly used for nearest neighbor (NN) and similarity search in applications involving high-dimensional data and/or costly distance computations. Today, many such applications require a proof of result correctness. In this setting, clients issue NN queries to a server that maintains a database signed by a trusted authority. The server returns the NN set along with supplementary information that permits result verification using the dataset signature. Unfortunately, an adaptation of the multi-step NN algorithm incurs prohibitive network overhead due to the transmission of false hits, i.e., records that are not in the NN set, but are nevertheless necessary for its verification. In order to alleviate this problem, we present a novel technique that reduces the size of each false hit. Moreover, we generalize our solution for a distributed setting, where the database is horizontally partitioned over several servers. Finally, we demonstrate the effectiveness of the proposed solutions with real datasets of various dimensionalities.