Numerical Method for Comparison on Homomorphically Encrypted Numbers

We propose a new method to compare numbers which are encrypted by Homomorphic Encryption (HE). Previously, comparison and min/max functions were evaluated using Boolean functions where input numbers are encrypted bit-wise. However, the bit-wise encryption methods require relatively expensive computations for basic arithmetic operations such as addition and multiplication. In this paper, we introduce iterative algorithms that approximately compute the min/max and comparison operations of several numbers which are encrypted word-wise. From the concrete error analyses, we show that our min/max and comparison algorithms have Theta(alpha) and Theta(alpha log alpha) computational complexity to obtain approximate values within an error rate 2(-alpha), while the previous minimax polynomial approximation method requires the exponential complexity Theta(2(alpha/2)) and Theta(root alpha . 2(alpha/2)), respectively. Our algorithms achieve (quasi-)optimality in terms of asymptotic computational complexity among polynomial approximations for min/max and comparison operations. The comparison algorithm is extended to several applications such as computing the top-k elements and counting numbers over the threshold in encrypted state. Our method enables word-wise HEs to enjoy comparable performance in practice with bit-wise HEs for comparison operations while showing much better performance on polynomial operations. Computing an approximate maximum value of any two l-bit integers encrypted by HEAAN, up to error 2(l-10), takes only 1.14 ms in amortized running time, which is comparable to the result based on bit-wise HEs.