ComBo: A Novel Functional Bootstrapping Method for Efficient Evaluation of Nonlinear Functions in the Encrypted Domain.

The application of Fully Homomorphic Encryption (FHE) to privacy issues arising in inference or training of neural networks has been actively researched over the last few years. Yet, although practical performances have been demonstrated on certain classes of neural networks, the inherent high computational cost of FHE operators has prevented the scaling capabilities of FHE-based encrypted domain inference to the large and deep networks used to deliver advanced classification functions such as image interpretation tasks. To achieve this goal, a new hope is coming from TFHE functional bootstrapping which, rather than being just used for refreshing ciphertexts (i.e., reducing their noise level), can be used to evaluate operators which are difficult to express as low complexity arithmetic circuits, at no additional cost. In this work, we first propose ComBo (Composition of Bootstrappings) a new full domain functional bootstrapping method with TFHE for evaluating any function of domain and codomain the real torus T by using a small number of bootstrappings. This result improves on previous approaches: like them, we allow for evaluating any functions, but with error rates reduced by a factor of up to 2 80 . This claim is supported by a theoretical analysis of the error rate of other functional bootstrapping methods from the literature. The paper is concluded by extensive experimental results demonstrating that our method achieves better performances in terms of both time and precision, in particular for the Rectified Linear Unit (ReLU) function, a nonlinear activation function commonly used in neural networks. As such, this work provides a fundamental building-block towards scaling the homomorphic evaluation of neural networks over encrypted data.