In this study, we propose an efficient sorting method for encrypted data using fully homomorphic encryption (FHE). The proposed method extends the existing 2-way sorting method by applying the k-way sorting network for any prime k to reduce the depth in terms of comparison operation from O(log22 n) to O(klogk2 n), thereby improving performance for k slightly larger than 2, such as k=5. We apply this method to approximate FHE which is widely used due to its efficiency of homomorphic arithmetic operations. In order to build up the k-way sorting network, the k-sorter, which sorts k-numbers with a minimal comparison depth, is used as a building block. The approximate homomorphic comparison, which is the only type of comparison working on approximate FHE, cannot be used for the construction of the k-sorter as it is because the result of the comparison is not binary, unlike the comparison in conventional bit-wise FHEs. To overcome this problem, we propose an efficient k-sorter construction utilizing the features of approximate homomorphic comparison. Also, we propose an efficient construction of a k-way sorting network using cryptographic SIMD operations. To use the proposed method most efficiently, we propose an estimation formula that finds the appropriate k that is expected to reduce the total time cost when the parameters of the approximating comparisons and the performance of the operations provided by the approximate FHE are given. We also show the implementation results of the proposed method, and it shows that sorting 56 = 15625 data using 5-way sorting network can be about 23.3% faster than sorting 214 = 16384 data using 2-way.

中文翻译：

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利用 k-Way 排序网络对同态加密数据进行高效排序


在本研究中，我们提出了一种使用完全同态加密（FHE）的加密数据的有效排序方法。该方法扩展了现有的2路排序方法，通过对任意素数k应用k路排序网络，将比较操作的深度从O(log22 n)减少到O(klogk2 n)，从而提高k的性能略大于2，例如k=5。我们应用这种方法来近似 FHE，由于其同态算术运算的效率而被广泛使用。为了构建 k 路排序网络，使用 k 排序器作为构建块，该 k 排序器以最小比较深度对 k 个数字进行排序。近似同态比较是唯一适用于近似 FHE 的比较类型，它不能用于构建 k 排序器，因为比较的结果不是二进制的，这与传统按位 FHE 中的比较不同。为了克服这个问题，我们利用近似同态比较的特征提出了一种有效的 k 排序器结构。此外，我们提出使用加密 SIMD 操作有效构建 k 路排序网络。为了最有效地使用所提出的方法，我们提出了一个估计公式，当给定近似比较的参数和近似 FHE 提供的操作性能时，该估计公式可以找到适当的 k，预计可以减少总时间成本。我们还展示了该方法的实现结果，结果表明使用 5 路排序网络对 56 = 15625 数据进行排序可以比使用 2 路排序网络对 214 = 16384 数据进行排序快约 23.3%。