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There is an encryption scheme where the votes are encrypted with ElGamal and the decryption key is the secret that is shared among the authorities. After everybody voted they publish their part of the secret according to the Shamir tresholding scheme and everybody can calculate the tally.

Is it possible that every voter is also an authority and that a threshold scheme is implemented such that the secret is if and only if reveled when a relative majority publishes their parts of the secret?

The basic idea of the voting scheme is from Cramer, Genaro and Shoemaker

https://www.win.tue.nl/~berry/papers/euro97.pdf

the requirements are:

  1. eligible voters can cast their votes
  2. only eligible voters are allowed to cast votes
  3. Privacy: the secrecy of an individual vote must be maintained
  4. Correctness: the votes must be counted correctly
  5. Robustness: the scheme must work correctly even in the presence of attackers and dishonestly involved parties
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  • $\begingroup$ Do you care about votes being confidential? Please reference the specific voting scheme you're referring to, because you've leaving too many requirements unknown. Recent questions on this topic have been about schemes which shuffle the votes to maintain confidentiality. $\endgroup$
    – knaccc
    Jan 18 at 16:01
  • $\begingroup$ @knaccc updated $\endgroup$
    – New2Math
    Jan 19 at 2:43
  • $\begingroup$ It's to be understood that "eligible" voters are those allowed to vote, not those who can be elected. At least to my french ears, that was non-obvious at first. $\endgroup$
    – fgrieu
    Jan 19 at 6:56

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The scheme you've referenced protects the confidentiality of individual votes only if you trust a threshold of authorities to only collectively agree to decrypt the aggregate of the individual votes. There is nothing to stop a threshold of authorities from violating this trust and colluding to decrypt individual votes (the authorities can only knowingly collude to do so).

There is nothing to prevent the voters from also being the authorities. You would have to have faith that a threshold of voters would not collude to reveal individual votes. You may decide this risk of a large number of individual voters colluding to violate individual vote privacy is lower than the risk of a small number of authorities colluding to violate individual vote privacy.

Note that unless you have a system for determining voter eligibility that can be verified by voters and not by a traditional trusted authority, voters acting as authorities would have to trust that a traditional authority has correctly certified as eligible the set of encrypted votes which will be collectively counted by the voter-authorities.

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