Homomorphic secret sharing

In cryptography, homomorphic secret sharing is a type of secret sharing algorithm in which the secret is encrypted via homomorphic encryption. A homomorphism is a transformation from one algebraic structure into another of the same type so that the structure is preserved. Importantly, this means that for every kind of manipulation of the original data, there is a corresponding manipulation of the transformed data.
Technique.
Homomorphic secret sharing is used to transmit a secret to several recipients as follows:
Examples.
Suppose a community wants to perform an election, using a decentralized voting protocol, but they want to ensure that the vote-counters won't lie about the results. Using a type of homomorphic secret sharing known as Shamir's secret sharing, each member of the community can add their vote to a form that is split into pieces, each piece is then submitted to a different vote-counter. The pieces are designed so that the vote-counters can't predict how any alterations to each piece will affect the whole, thus, discouraging vote-counters from tampering with their pieces. When all votes have been received, the vote-counters combine them, allowing them to recover the aggregate election results.
In detail, suppose we have an election with:
Features.
This protocol works as long as not all of the formula_1 authorities are corrupt — if they were, then they could collaborate to reconstruct formula_2 for each voter and also subsequently alter the votes.
The protocol requires t+1 authorities to be completed, therefore in case there are N>t+1 authorities, N-t-1 authorities can be corrupted, which gives the protocol a certain degree of robustness.
The protocol manages the IDs of the voters (the IDs were submitted with the ballots) and therefore can verify that only legitimate voters have voted.
Under the assumptions on t:
The protocol implicitly prevents corruption of ballots.
This is because the authorities have no incentive to change the ballot since each authority has only a share of the ballot and has no knowledge how changing this share will affect the outcome.