Homomorphic Comparison Method Based on Dynamically Polynomial Composite Approximating Sign Function

Performing comparison operations between two numbers is one of the most commonly used operations in some applications. However, performing non-arithmetic operations, such as comparison, on data encrypted with fully homomorphic encryption that only supports addition and multiplication operations, has always been a challenging task. Recently, Cheon et al.(Asiacrypt 2020) proposed a new comparison method in homomorphic encryption using composite polynomial approximation of the sign function, and proved that their method has optimal asymptotic complexity. In this paper, we propose a new comparison method using dynamic polynomial composition to approximate the sign function. We analyze the optimality of the depth consumption and computational complexity of this approximation method. Additionally, we extend this comparison algorithm to multiple applications, such as Min/Max algorithm and k-way sorting network algorithm. Our numerical analysis shows that for computing the approximate comparison of any two 32-bit integers encrypted by HEAAN, our homomorphic comparison algorithm (for $\epsilon=2^{-16},\alpha=2^{-32}$) requires 0.83ms in amortized running time. This reduces the running time by about 25% compared to the previous optimal method.