This allows calculating the tally within the ciphertext space by simply multiplying the ciphertexts. Typically, the ballot ciphertexts have a vector form, and are multiplied component-wise. For example, imagine there are 4 delegates, and we have six votes, which we represent below as row vectors.
Each component of the above row vectors encrypts the value ‘0’ or ‘1’. These ciphertexts are multiplied together to give the tally result, which is still in encrypted form. The result can then be decrypted without compromising vote secrecy for individual ballots, which are never encrypted.
Going back to the scheme, this tally result after the first ‘+’ operation gives us the Delegate totals, which is still double bordered. The next step is to combine these delegate totals with the publicly expressed votes that the delegates make.
which would then be processed further together with the direct overriding votes.
but with the new approach we don’t know how many votes each delegate received. We only know that delegate 1 and 4 summing 1, whereas 2 and 3 summing 5, but not more information about them individually. Again, this was because we delayed decryption until the last possible step, and performed more operations in the ciphertext space.
The rest of the scheme follows the same pattern as the Mixnet/Mixnet scheme. At this point we have a partial tally corresponding to the ‘Weight for choices’. We must also take into account those voters that chose to override their delegate’s choice casting a direct vote. These direct encrypted ballots are processed by a mixnet to give us a partial tally corresponding to direct choices. The partial tallies corresponding to weighted delegate votes and direct votes are summed together to give the final tally, which is the election result. Like in Mixnet/Mixnet, we must invalidate duplicates by eliminating from the set of delegate ballots those that correspond to voters who voted directly (this is not shown in the diagram).
In summary, our second scheme supporting liquid democracy again inherits the privacy, and end-to-end verifiability properties of each of its two components, each of which appear in different voting schemes (typically referred to as “based on homomorphic encryption” and “based on mixnets”). There are differences, however. The use of homomorphic tallying for delegate votes yields better privacy for delegate totals, which are somewhat protected given that the weighting tally occurs in the ciphertext space. The exact level of privacy depends on the choices in the election: fewer options result in higher weight privacy, as there will be more delegates grouped into each choice.
We will revisit these considerations when we compare the three schemes at the end of the series.
 Aggelos Kiayias and Moti Yung. The Vector-Ballot E-Voting Approach.
 Aggelos Kiayias Michael Korman David Walluck. An Internet Voting System Supporting User Privacy.