On the privacy of federated Clustering: A Cryptographic View

The privacy concern in federated clustering has attracted considerable attention in past decades. Many privacy-preserving clustering algorithms leverage cryptographic techniques like homomorphic encryption or secure multiparty computation, to guarantee full privacy, i.e., no additional information is leaked other than the final output. However, given the iterative nature of clustering algorithms, consistently encrypting intermediate outputs, such as centroids, hampers efficiency. This paper delves into this intricate trade-off, questioning the necessity of continuous encryption in iterative algorithms. Using the federated K-means clustering as an example, we mathematically formulate the problem of reconstructing input private data from the intermediate centroids as a classical cryptographic problem called hidden subset sum problem (HSSP)-extended from an NP-complete problem called subset sum problem (SSP). Through an in-depth analysis, we show that existing lattice-based HSSP attacks fail in reconstructing the private data given the knowledge of intermediate centroids, thus it is secure to reveal them for the sake of efficiency. To the best of our knowledge, our work is the first to cast federated clustering's privacy concerns as a cryptographic problem HSSP such that a concrete and rigorous analysis can be conducted.