Efficient Homomorphic Evaluation of Arbitrary Uni/Bivariate Integer Functions and Their Applications

We propose how to homomorphically evaluate arbitrary univariate and bivariate integer functions such as division. A prior work proposed by Okada et al. (WISTP'18) uses polynomial evaluations such that the scheme is still compatible with the SIMD operations in BFV and BGV schemes, and is implemented with the input domain 𝕫257. However, the scheme of Okada et al. requires the quadratic numbers of plaintext-ciphertext multiplications and ciphertext-ciphertext additions in the input domain size, and although these operations are more lightweight than the ciphertext-ciphertext multiplication, the quadratic complexity makes handling larger inputs quite inefficient. In this work, first we improve the prior work and also propose a new approach that exploits the packing method to handle the larger input domain size instead of enabling the SIMD operation, thus making it possible to work with the larger input domain size, e.g., 𝕫215 in a reasonably efficient way. In addition, we show how to slightly extend the input domain size to 𝕫216 with a relatively moderate overhead. Further we show another approach to handling the larger input domain size by using two ciphertexts to encrypt one integer plaintext and applying our techniques for uni/bivariate function evaluation. We implement the prior work of Okada et al., our improved version of Okada et al., and our new scheme in PALISADE with the input domain 𝕫215, and confirm that the estimated run-times of the prior work and our improved version of the prior work are still about 117 days and 59 days respectively while our new scheme can be computed in 307 seconds.