An Efficient Fully Homomorphic Encryption Sorting Algorithm Using Addition Over TFHE

Fully homomorphic encryption (FHE) can effectively protect data privacy. It allows users to entrust data operations to a third party on a cloud server without revealing their own privacy. Sorting algorithms are the fundamental technique of managing and processing data. Currently, the sorting algorithms based on the traditional FHE schemes of BGV (Brakerski-Gentry-Vaikuntanathan), BFV (BrakerskiFan-Vercauteren) and CKKS (Cheon-Kim-Kim-Song) are difficult to implement and are inefficient because of their slow bootstrapping techniques. TFHE (Fast Fully Homomorphic Encryption over the Torus), a fast FHE scheme over the torus based on GSW (Gentry-Sahai-Waters), significantly decrease the time cost of bootstrapping. In this paper, we first design the bitwise FHE comparison and swap operations using the bootstrapped binary gates of the TFHE library. With the two operations, a bubble sort algorithm is proposed. By reducing the depth of sorting circuits in the proposed bubble sort, we further present AdditionSort, an efficient sorting algorithm using homomorphic addition, which can support the operations of arbitrary array length and unlimited element size. The experiments show that AdditionSort is nearly 50% faster than the bubble sort when the size of an element exceeds 32 bits.