Revisiting the functional bootstrap in TFHE

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.

Survey on Fully Homomorphic Encryption, Theory, and Applications

“…This construction can also be used to evaluate homomorphically several functions over the same encrypted message simultaneously. In a recent work, Guimarães, Borin and Aranha [115] optimized the bootstrapping procedure to evaluate multiple functions on large ciphertexts and Chillotti, Ligier, Orfila and Tap [113] proposed several enhancements, including a generalized method to evaluate several functions at once without adding additional error. Finally, in [116], Chen, Chillotti and Song proposed a multi-key homomophic encryption scheme from TFHE.…”

Survey on Fully Homomorphic Encryption, Theory, and Applications

“…This construction can also be used to evaluate homomorphically several functions over the same encrypted message simultaneously. In a recent work, Guimarães, Borin and Aranha [125] optimized the bootstrapping procedure to evaluate multiple functions on large ciphertexts and Chillotti, Ligier, Orfila and Tap [117] proposed several enhancements, including a generalized method to evaluate several functions at once without adding additional error. Finally, in [126], Chen, Chillotti and Song proposed a multi-key homomophic encryption scheme from TFHE.…”

Survey on Fully Homomorphic Encryption, Theory and Applications

“…This construction can also be used to evaluate homomorphically several functions over the same encrypted message simultaneously. In a recent work, Guimarães, Borin and Aranha [125] optimized the bootstrapping procedure to evaluate multiple functions on large ciphertexts and Chillotti, Ligier, Orfila and Tap [121] proposed several enhancements, including a generalized method to evaluate several functions at once without adding additional error. Finally, in [126], Chen, Chillotti and Song proposed a multi-key homomophic encryption scheme from TFHE.…”