Fully Homomorphic Encryption (FHE) is a corpus of cryptographic techniques that allow to compute directly over encrypted data. Since its inception around 15 years ago, FHE has been the subject of a lot of research towards more efficiency and better practicality. From a security perspective, however, FHE still raises a number of questions and challenges. In particular, all the FHE used in practice, mainly BFV, BGV, CKKS and TFHE, achieve only CPA-security, which is sometimes referred to as security against passive adversaries.

Over the last few years, a number of works have investigated the security of FHE in the beyond-CPA regime with new security notions (CPAD, FuncCPA, vCCA, vCCAD, and others) being proposed and studied, leading to new attacks and constructions and, overall, a better understanding of FHE security in that regime.

With respect to CCA security, recent works (2024) have defined new security notions, which are stronger than CCA1 and shown to be achievable by both exact and approximate FHE schemes. Leveraging on these advances, the present thesis will aim to design practical FHE-style malleable schemes enforcing CCA security properties, at least for specific applications.