Overflow-Detectable Floating-Point Fully Homomorphic Encryption

Abstract

A floating-point fully homomorphic encryption (FPFHE) is proposed, which is based on torus fully homomorphic encryption equipped with programmable bootstrapping. Specifically, FPFHE for 32-bit and 64-bit floating-point messages are implemented, the latter showing the state-of-the-art precision among FHEs. Also, a ciphertext is constructed for checking if an overflow has occurred or not while evaluating arithmetic circuits with the proposed FPFHE, which is useful when the message space or arithmetic circuit is too complex to estimate a bound of outputs such as some deep learning applications. Also, homomorphic algorithms, which are crucial components of overflow detectable (OD)-FPFHE, are constructed. First, a state-of-the-art bootstrapping method of TFHE is extended to bootstrap larger messages by using NTT-friendly integer modulus. Second, a subgaussian analysis method is proposed without assuming independent heuristic on AP/GINX-bootstrapping even if the deterministic gadget decomposition is used. Third, the blind rotation algorithm of TFHE is modified such that any secret key having finite non-zero values can be used while keeping the number of NTT operations the same as when the binary key is used. Fourth, various homomorphic algorithms are proposed such as evaluating min and max, lifting a constant message to the monomial exponent, counting the number of consecutive zeros from the most significant in the fraction, and performing carryover after homomorphic operation of floating-point numbers. Finally, 32-bit and 64-bit OD-FPFHEs are implemented and simulation results are provided to confirm that they work well even for extreme cases. Also, it is verified that homomorphic overflow detection is well-operated.

Keywords

• Fully homomorphic encryption
• homomorphic floating-point arithmetic
• homomorphic overflow detection
• subgaussian error analysis