Homomorphic sorting is an operation that blindly sorts a given set of encrypted numbers without decrypting them (thus, there is no need for the secret key). In this article, we propose a new, efficient, and scalable method for homomorphic sorting of numbers: polynomial rank sort algorithm. To put the new algorithm in a comparative perspective, we provide an extensive survey of classical sorting algorithms and networks that are not directly suitable for homomorphic computation. We also include, in our discussions, two of our previous algorithms specifically designed for homomorphic sorting operation: direct and greedy sort, and explain how they evolve from classical sorting networks. We theoretically show that the new algorithm is superior in terms of multiplicative depth when compared with all other algorithms. When batched implementation is used, the number of comparisons is reduced from $\mathcal {O}(N^2)$O(N2) to $\mathcal {O}(N)$O(N) provided that the number of slots is larger than or equal to the number of elements in the set. Our software implementation results confirm that the new algorithm is several orders of magnitude faster than many methods in the literature. Also, the polynomial sort algorithm scales better than the fastest algorithm in the literature to the best our knowledge although for small sets the execution times are comparable. The proposed algorithm is amenable to parallel implementation as most time consuming operations in the algorithm can naturally be performed concurrently.

中文翻译：

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具有更好可扩展性的同态排序

同态排序是一种对一组给定的加密数字进行盲目排序而不解密它们的操作（因此，不需要密钥）。在本文中，我们提出了一种新的、高效且可扩展的数字同态排序方法：多项式排序算法。为了将新算法放在比较的角度，我们对不直接适用于同态计算的经典排序算法和网络进行了广泛的调查。在我们的讨论中，我们还包括我们之前专门为同态排序操作设计的两种算法：直接贪婪排序，并解释它们如何从经典排序网络演变而来。我们从理论上表明，与所有其他算法相比，新算法在乘法深度方面更胜一筹。使用批处理实现时，比较次数从$\mathcal {O}(N^2)$哦(N2) 至 $\mathcal {O}(N)$哦(N)前提是槽的数量大于或等于集合中的元素数量。我们的软件实现结果证实，新算法比文献中的许多方法快几个数量级。此外，多项式排序算法比文献中最快的算法更好地扩展到我们的知识范围，尽管对于小集合，执行时间是可比的。所提出的算法适合并行实现，因为算法中最耗时的操作可以自然地并发执行。