I am reading the Wikipedia article https://en.wikipedia.org/wiki/Homomorphic_encryption and it lists unpadded RSA, ElGamal, Goldwasser-Micali, Benaloh, Paillier as possible partially homomorphic encryption (PHE) schemes.
Of these, only Goldwasser-Micali seems to be for bits and it is XOR-homomorphic. The other encryption schemes seem to be for group elements, and it seems like if you say you only encrypt 0 or 1 then the operations won't hold (because, e.g., $1+1 =2$ now).
Assuming that I only want to encrypt 0 or 1, is there a partial HE that can do ANDs (aka multiplication)? Or, is it only possible to do multiplication on a larger group? I understand that many fully homomorphic encryption schemes do just work on bits, but then there's issues with noise growing/boot strapping. (Whereas, for example, multiplying two ElGamal ciphertexts together is a "clean" operation).
Is there a simple PHE that works on bits only and allows you to do an unbounded number of ANDs? Or is FHE required?
To try to give some context: what if there are $n$ people who are voting yes/no with 1 or 0 and you want to find out if the vote is unanimous for yes? I guess here there's some issue with having to do some sort of threshold decryption or multi-key encryption as well, and from my understanding from 2016/196 or 2017/956that makes the ciphertext be of size $n$ also, which seems like a big hassle for a simple operation. That's why I thought perhaps a PHE would be enough, and perhaps give you the chance of having smaller ciphertexts.