Sorting method for fully homomorphic encrypted data using the cryptographic single-instruction multiple-data operation

In this paper, we present a faster (wall-clock time) sorting method for numerical data subjected to fully homomorphic encryption (FHE). Owing to circuit-based construction and the FHE security property, most existing sorting methods cannot be applied to encrypted data without significantly compromising efficiency. The proposed algorithm utilizes the cryptographic single-instruction multiple-data (SIMD) operation, which issupported by most existing FHE algorithms, to reduce the computational overhead. We conducted a careful analysis of the number of required recryption operations, which are the computationally dominant operations in FHE. Accordingly, we verified that the proposed SIMD-based sorting algorithm completes the given task more quickly than existing sorting methods if the number of data items and (or) the maximum bit length of each data item exceed specific thresholds.