Accelerating Sorting of Fully Homomorphic Encrypted Data

Abstract

Sorting is an age old problem in Computer Science. Recently with the advent of cloud computing this problem is revisited on encrypted data. This paper tries to evaluate the possibility of applying the recently discovered Fully Homomorphic Encryption schemes to sort encrypted text. The paper first develops fully homomorphic circuits for performing comparison based swaps and then employs them to realize conventional sorting algorithms. Since the sorting time grows exponentially with the input size, it is required to propose suitable measures to reduce it; the delay occuring due to the costly Recrypt operation which removes the noise in the Homomorphic computations. The paper then investigates the opportunity of reducing Recrypt by experimenting on the average errors introduced due to wrong comparisons, which arise due to the removal of the de-noising step. Results show that suitably choosing the number of Recrypt operations results in an almost sorted array. This motivates to develop a two-stage sorting called LazySort: the first phase performing a Bubble sort with reduced Recrypt operations to result in an almost sorted array, to be followed by a second stage which employs an Insertion sort with all Recrypt operations. Detailed experiments show that helps to obtain a significant speed up in the sorting time.