Suppose there are three voters $P$, $Q$ and $R$, and each will vote only on one candidate out of $X$, $Y$ or $Z$, with a 6 bit vote vector corresponding to $X $, $Y$ and $Z$ respectively (with 2 bits each for a candidate. E.g., $( 01, 00, 00 )$ signifies a vote for $X$).
How can any voter find out the total number of votes cast for each candidate without disclosing their preferences by the use of secure multiparty computation?
Each 6 bit vote vector is encrypted and sent to the some third party who will then take the product of ciphertext votes and then decrypt the product to yield a bit vector which shows the total number of preferences received by each candidate (Paillier cryptosystem).
What homomorphic property must be held by the cryptosystem in the plaintext and in the ciphertext space to satisfy the objective?
Finally, I need some theoretical explanation finding secure sum using SMP (with and without random seed) and SMC to sort the numbers.