Abstract
The FHEW cryptosystem introduced the idea that an arbitrary function can be evaluated within the bootstrap procedure as a table lookup. The faster bootstraps of TFHE strengthened this approach, which was later named Functional Bootstrap (Boura et al., CSCML’19). From then on, little effort has been made towards defining efficient ways of using it to implement functions with high precision. In this paper, we introduce two methods to combine multiple functional bootstraps to accelerate the evaluation of reasonably large look-up tables and highly precise functions. We thoroughly analyze and experimentally validate the error propagation in both methods, as well as in the functional bootstrap itself. We leverage the multi-value bootstrap of Carpov et al. (CT-RSA’19) to accelerate (single) lookup table evaluation, and we improve it by lowering the complexity of its error variance growth from quadratic to linear in the value of the output base. Compared to previous literature using TFHE’s functional bootstrap, our methods are up to 2.49 times faster than the lookup table evaluation of Carpov et al. (CT-RSA’19) and up to 3.19 times faster than the 32-bit integer comparison of Bourse et al. (CT-RSA’20). Compared to works using logic gates, we achieved speedups of up to 6.98, 8.74, and 3.55 times over 8-bit implementations of the functions ReLU, Addition, and Maximum, respectively.
Highlights
The efficient evaluation of non-linear functions with high precision is a challenge for homomorphic encryption schemes and many of them rely on arithmetic approximations, such as Taylor, Fourier, and Chebyshev series [BGGJ19, CCS19, KWN20]
As we described in Section 2.3.2, the multi-value bootstrap relies on multiplications between the accumulator (ACC in line 4 of Algorithm 5) and the lookup table (LUT) encoded as polynomials in ZN [X]
Lookup tables are very commonly used in homomorphic circuits, and the implementation of Carpov et al [CIM19] is one of the most recent and efficient of them. It introduces the multi-value bootstrap of TFHE, which we explore in our tree-based method
Summary
The efficient evaluation of non-linear functions with high precision is a challenge for homomorphic encryption schemes and many of them rely on arithmetic approximations, such as Taylor, Fourier, and Chebyshev series [BGGJ19, CCS19, KWN20]. They allow them to work with packed messages in a SIMD1 manner [BGH13], which greatly reduces the amortized cost of operations. All currently known fully homomorphic encryption (FHE) schemes rely on noisy ciphertexts for security, i.e., the encryption process adds a small error (noise) to the message This error grows when performing arithmetic, and, eventually, it might affect
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