Fully Homomorphic Encryption (FHE) schemes enable to compute over encrypted data. Among them, TFHE [CGGI17] has the great advantage of offering an efficient method for bootstrapping noisy ciphertexts, i.e., reduce the noise. Indeed, homomorphic computation increases the noise in ciphertexts and might compromise the encrypted message. TFHE bootstrapping, in addition to reducing the noise, also evaluates (for free) univariate functions expressed as look-up tables. It however requires to have the most significant bit of the plaintext to be known a priori, resulting in the loss of one bit of space to store messages. Furthermore it represents a non negligible overhead in terms of computation in many use cases.In this paper, we propose a solution to overcome this limitation, that we call Programmable Bootstrapping Without Padding (WoP-PBS). This approach relies on two building blocks. The first one is the multiplication à la BFV [FV12] that we incorporate into TFHE. This is possible thanks to a thorough noise analysis showing that correct multiplications can be computed using practical TFHE parameters. The second building block is the generalization of TFHE bootstrapping introduced in this paper. It offers the flexibility to select any chunk of bits in an encrypted plaintext during a bootstrap. It also enables to evaluate many LUTs at the same time when working with small enough precision. All these improvements are particularly helpful in some applications such as the evaluation of Boolean circuits (where a bootstrap is no longer required in each evaluated gate) and, more generally, in the efficient evaluation of arithmetic circuits even with large integers. Those results improve TFHE circuit bootstrapping as well. Moreover, we show that bootstrapping large precision integers is now possible using much smaller parameters than those obtained by scaling TFHE ones.

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Classification algorithms/tools become more and more powerful and pervasive. Yet, for some use cases, it is necessary to be able to protect data privacy while benefiting from the functionalities they provide. Among the tools that may be used to ensure such privacy, we are focusing in this paper on functional encryption. These relatively new cryptographic primitives enable the evaluation of functions over encrypted inputs, outputting cleartext results. Theoretically, this property makes them well-suited to process classification over encrypted data in a privacy by design’ rationale, enabling to perform the classification algorithm over encrypted inputs (i.e. without knowing the inputs) while only getting the input classes as a result in the clear.In this paper, we study the security and privacy issues of classifiers using today practical functional encryption schemes. We provide an analysis of the information leakage about the input data that are processed in the encrypted domain with state-of-the-art functional encryption schemes. This study, based on experiments ran on MNIST and Census Income datasets, shows that neural networks are able to partially recover information that should have been kept secret. Hence, great care should be taken when using the currently available functional encryption schemes to build privacy-preserving classification services. It should be emphasized that this work does not attack the cryptographic security of functional encryption schemes, it rather warns the community against the fact that they should be used with caution for some use cases and that the current state-ofthe-art may lead to some operational weaknesses that could be mitigated in the future once more powerful functional encryption schemes are available.

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