Vers une arithmétique homomorphe pratique et efficace

Fully Homomorphic Encryption (FHE) is a family of encryption schemes permitting operations over encrypted data.Thanks to this property, FHE schemes allow the evaluation of circuits while preserving user privacy.As a result, FHE has applications in diverse domains such as machine learning, blockchain, and so on.Over the last decades, the field has evolved from schemes taking an impracticable amount of time to evaluate small circuits to schemes able to evaluate complex circuits in a reasonable amount of time, leading to the beginning of the adoption of FHE at an industrial level.This thesis begins in this context, with a focus on the TFHE scheme.TFHE is an FHE scheme that is highly efficient at performing operations over small message precision, mainly due to a core operation called bootstrapping, which is used all along the circuit to ensure correctness.But despite its efficiency, TFHE still has some limitations that we aim to overcome in this manuscript.As mentioned before, TFHE has very efficient algorithms to work over small message precision, but many circuits work with primitive data types such as 32- or 64-bit integers or floating-point numbers.Regarding this first limitation, we study how to efficiently represent these data types with TFHE by using dedicated encodings and algorithms.Although TFHE is one of the fastest FHE schemes, it remains slow compared to plaintext operations.To reduce the timing differences, we study low-level algorithms and new primitives to reduce the global cost of FHE operations.All these improvements permit the creation of efficient and practical homomorphic arithmetic, bridging the gap between the clear world and the homomorphic world.