Equality checking using additive homomorphic encryption Given two ciphertexts $c_1 = enc(p_1)$ and $c_2= enc(p_2)$ using any additive homomorphic encryption scheme (or specifically Paillier). 

Can we find out whether the underlying plaintexts $p_1,p_2$ are equal without decrypting $c_1,c_2$ i.e. homomorphically?
I'm not sure if I understand what you are asking, so I'll clarify what I am about to answer. We are given two ciphertexts and we want to know if they encrypt the same plaintext or if they encrypt different plaintexts, and we want to do this without revealing anything but this fact.

Then, using the additive homomorphism, it's possible to compute $c=enc(r\cdot (p_1-p_2))$. If $p_1=p_2$ then this is an encryption of 0; else it's an encryption of a random value.

Now, let's consider the setting where one party holds the private key and the other has $c_1,c_2$. Then, the 2nd party can compute the above as I showed and send it to the first party. The first party can then decrypt and see if it is 0 or not. There exist ways for the first party to prove this to the second party efficiently.

I hope that this answers the question.