International Association
for Cryptologic Research

Sorting arrays encrypted under fully homomorphic encryption remains a fundamental challenge due to the high cost of private comparisons and the incompatibility of conventional sorting algorithms with the encrypted domain. Recently, Hong et al. (IEEE Transactions on Information Forensics and Security, 2021) proposed a $k$-way sorting network tailored to encrypted real numbers, but its reliance on multiple comparison stages incurs substantial multiplicative depth and significant bootstrapping overhead, even for modest array sizes.

In this work, we propose a novel rank-based sorting algorithm for encrypted real numbers that performs only a single comparison stage, thereby eliminating the need for bootstrapping operations. Our empirical evaluation demonstrates that the proposed method significantly outperforms the $k$-way approach for small to medium array sizes $(n\leq 1024)$, achieving a $46.91\times$ speedup at $n=256$ with a total runtime of $79$ seconds. Furthermore, we examine the recent matrix-based rank sort method by Mazzone et al. (USENIX Security '25) and show that integrating our optimized rank construction improves its efficiency. Specifically, we achieve $1.77\times$ and $2.43\times$ performance gains for $n=128$ and $n=512$, respectively.