What's the difficulty of using elliptic curves to design homomorphic encryption protocols? I have recently been very interested in elliptic curves because they are a powerful tool in crypto, ECC, pairing, etc. However, it seems that elliptic curves are not popular in homomorphic encryption. I understand that lattice-based crypto is currently the only way to construct FHE, like LWE. Elliptic curves can only be used to construct PHE; for example, BGN. After doing some research as a freshman in HE, I found that there is not much effort in trying to use EC to construct FE. So, I'm just wondering why. Is it because elliptic curves have some issues in their structure, making it hard to construct HE? Or is it because lattice-based construction is a hot topic, so researchers prefer it over EC?
From my point of view, Homomorphism is a inherent attribute of certain lattice-based scheme.Researchers try every means of solution to enhance the attribute,but not all primitives have this attribute.