Computation on ciphertexts of all known fully homomorphic encryption (FHE) schemes induces some noise, which, if too large, will destroy the plaintext. Therefore, the bootstrapping technique that re-encrypts a ciphertext and reduces the noise level remains the only known way of building FHE schemes for arbitrary unbounded computations. The bootstrapping step is also the major efficiency bottleneck in current FHE schemes. A promising direction towards improving concrete efficiency is to exploit the bootstrapping process to perform useful computation while reducing the noise at the same time. We show a bootstrapping algorithm, which embeds a lookup table and evaluates arbitrary functions of the plaintext while reducing the noise. Depending on the choice of parameters, the resulting homomorphic encryption scheme may be either an exact FHE or homomorphic encryption for approximate arithmetic. Since we can evaluate arbitrary functions over the plaintext space, we can use the natural homomorphism of Regev encryption to compute affine functions without bootstrapping almost for free. Consequently, our algorithms are particularly suitable for circuits with many additions and scalar multiplication gates. We achieve record speeds for such circuits. For example, in the exact FHE setting, we achieve a speedup of a factor of over 3000x over state-of-the-art methods. Effectively, we bring the evaluation time from weeks or days down to a few hours or minutes. Furthermore, we note that the speedup gets more significant with the size of the affine function. We provide a tight error analysis and show several parameter sets for our bootstrapping. Finally, we implement our algorithm and provide extensive tests. We demonstrate our algorithms by evaluating different neural networks in several parameter and accuracy settings.
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