- Is that correct that Somewhat Homomorphic Encryption is more efficient that “Fully Homomorphic Encryption” (FHE) but less efficient than Partially Homomorphic Encryption (e.g Paillier encryption)?
- Is that correct that in Somewhat Homomorphic Encryption (e.g. used in Private Database Queries Using Somewhat Homomorphic Encryption (PDF)), the plaintext can be defined over a “known finite field”, so after homomorphic operations – when we decrypt – the plaintext is in that finite field?
PHE (partially homomorphic encryption) schemes are in general more efficient than SHE and FHE, mainly because they are homomorphic w.r.t to only one type of operation: addition or multiplication.
SHE is more general than PHE in the sense that it supports homomorphic operations with additions and multiplications. The drawback is that you can perform only a limited number hom. operations.
To answer your first question, a SHE scheme is also a PHE. This implies that PHE's are at least as efficient as SHE's.
FHE allows you to do an unbounded number of hom. operations by bootstrapping. This step can only be done (currently) by evaluating the decryption circuit inside a SHE scheme.
AFAIK, all plaintext / ciphertext operations are done within a known modulus. The only unknown thing should be the secret key. See a construction example here: DGHV Scheme.