We examine the relationship between correctness definitions for Fully Homomorphic Encryption (FHE) and the associated security definitions. We show that reactive notions of correctness imply IND-CPA-D and sIND-CPA-D security. But that to obtain both IND-CPA-D and sIND-CPA-D security we need to use a randomized version of the evaluation procedure. Such randomized evaluation procedures cause problems in real life deployments of FHE solutions, so we then go on to show how one can de-randomize the evaluation procedure and still obtain sIND-CPA-D security in the random oracle model for the specific FHE scheme of TFHE.
Smart et al. (Mon,) studied this question.