Approximating Max Function in Fully Homomorphic Encryption

Abstract

This study focuses on efficiently finding the location of the maximum value for large-scale values encrypted by the CKKS (Cheon—Kim—Kim–Song) method. To find the maximum value, logM+1 comparison operations and logM rotation operations, and 2logM+3 additions and 2logM+1 multiplications are required. However, there is no known way to find a k-approximate maximum value, i.e., a value with the same most significant k-bits as the maximum value. In this study, when the value range of all data in each slot in the ciphertext is [0, 1], we propose a method for finding all slot positions of values whose most significant k-bits match the maximum value. The proposed method can find all slots from the input ciphertexts where their values have the same most significant k-bits as the maximum value by performing 2k comparison operations, (4k+2) multiplications, (6k+2klogM+3) additions, and 2klogM rotation operations. Through experiments and complexity analysis, we show that the proposed method is more efficient than the existing method of finding all locations where the k MSB is equal to the maximum value. The result of this can be applied to various privacy-preserving applications in various environments, such as IoT devices.