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What I'll describe works with any homomorphic scheme, whether multiplicative (Elgamal) or additive (Paillier; maybe exponential Elgamal or BGN depending), but I'll describe it with multiplicative. I …

Yes (and always). Given $\mathsf{Enc}(a)$ and $b$, you can compute $\mathsf{Enc}(a \cdot b^{-1} \bmod{n})$ by simply computing $\hat{b}=b^{-1} \bmod{n}$ and $Enc(a)^\hat{b} \bmod{n^2}$. Paillier enc …

To answer your second question, Paillier and other CPA-secure homomorphic encryption schemes cannot provide order-preserving encryption. The security of these schemes rely on using a random factor dur …

Elgamal can be made additive by encrypting $g^m$ instead of $m$ with traditional Elgamal for some generator $g$ (usually the same one used to generate the public key). This variant is sometimes called …

Using exponential Elgamal as the encryption function, Define the list of candidates: e.g., Alice, Bob, Carol Voters submit an encryption of their vote: e.g., to voter for Alice: $v=\langle\mathsf{En …

It seems you want to decrypt the final value without revealing the private key. First, if someone knows the private key, they can issue a very simple non-interactive zero knowledge proof that the plai …