Improved Integer-wise Homomorphic Comparison and Division based on Polynomial Evaluation

Fully homomorphic encryption (FHE) is a promising tool for privacy-preserving applications, and it enables us to perform homomorphic addition and multiplication on FHE ciphertexts without decrypting them. FHE has two types: one supporting the exact computation and the other supporting the approximate computation. Further the FHE schemes supporting the exact computation have two types, bit-wise FHE, which encrypts a plaintext bit by bit, and integer-wise FHE, which encrypts a plaintext as an integer. Both types of FHE are important depending on the types of computation we need to execute securely. In this work, we focus on integer-wise FHE, and propose improved methods for integer-wise homomorphic comparison and division operations. For a comparison operation, we propose a method that halves the number of necessary homomorphic multiplications by introducing an odd function as an interpolated polynomial to be evaluated, as opposed to the previous work of Narumanchi et al. (AINA ’17). For a division operation, as opposed to the previous work of Okada et al. (WISTP ’18), we propose a simple method to reduce the processing time by introducing an equality function based on Fermat’s little theorem without changing the multiplicative depth, and show the analysis of why this approach can achieve better efficiency in detail. In our homomorphic division, the number of interpolated polynomials is reduced by half, thus also achieving the reduction of the processing time of precomputations and the number of polynomials to be stored. We also implement our improved methods in HElib, which is one of popular FHE libraries using the BGV encryption. As a result, we show that, e.g., in the plaintext space , our homomorphic comparison with the Paterson-Stockmeyer method is faster by a factor of about 5.61 compared with Narumanchi et al. (AINA ’17) and our homomorphic division is faster by a factor of about 1.45 compared with Okada et al. (WISTP ’18).