Abstract
The Shell sort algorithm is one of the most practically effective in-place sorting algorithms. However, it is difficult to execute this algorithm with its intended running time complexity on data encrypted using fully homomorphic encryption (FHE), because the insertion sort in Shell sort has to be performed by considering the worst-case input data. In this paper, in order for the sorting algorithm to be used on the FHE data, we modify the Shell sort with an additional parameter α, allowing exponentially small sorting failure probability. For a gap sequence of powers of two, the modified Shell sort with input array length n is found to have the trade-off between the running time complexity of O(n3/2√α+loglogn) and the sorting failure probability of 2-α. Its running time complexity is close to the intended running time complexity of O(n3/2) and the sorting failure probability can be made very low with slightly increased running time. Further, the near-optimal window length of the modified Shell sort is also derived via convex optimization. The proposed analysis of the modified Shell sort is numerically confirmed by using randomly generated arrays. For the practical aspect, our modification can be applied to any gap sequence, and we show that Ciura’s gap sequence, which is known to have good practical performance, is also practically effective when our modified Shell sort is applied. We compare our modified Shell sort with other sorting algorithms with the FHE over the torus (TFHE) library, and it is shown that this modified Shell sort has the best performance in running time among in-place sorting algorithms on homomorphic encryption scheme.
Highlights
Homomorphic encryption (FHE) is an encryption scheme that provides encrypted data with an evaluation algorithm, which enables addition or multiplication of plaintext without decryption [1], [2]
We prove that the running time complexity of the modified Shell sort is determined to be O(n3/2√α + log log n) with an sorting failure probability (SFP) of 2−α for powers-of-two gap sequence, which consists of all powers of two less than the length of the array
Before analyzing the running time complexity and the sorting failure probability of the modified Shell sort with power-of-two gap sequence, we introduce the main idea of the analysis to help readers to understand the following theorems and lemmas
Summary
Homomorphic encryption (FHE) is an encryption scheme that provides encrypted data with an evaluation algorithm, which enables addition or multiplication of plaintext without decryption [1], [2]. With a fixed negligible SFP, we can set a small window length so that the running time is asymptotically faster than that of the naive version of the Shell sort on the FHE data. We propose a modified Shell sort with an additional parameter α on the FHE data, and derive its theoretical trade-off between the running time complexity O(n3/2√α + log log n) and the SFP 2−α when the gap sequence is power-of-two sequence. The numerical simulation with TFHE homomorphic encryption scheme is performed, and the modified Shell sort with Ciura’s gap sequence is proven to have the best running time performance in the practical situation among the in-place sorting algorithms for the FHE setting
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