Is there any standard solution to the colluding problem in Shamir's secret sharing i.e. How should I prevent my k friends from colluding and recovering my secret?
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1$\begingroup$ Use a $(k+1, n)$ threshold scheme? $\endgroup$– ponchoAug 22, 2022 at 20:22
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Well, if $k$ is the threshold, this is completely within the design of Shamir secret sharing, i.e., the scheme is working as intended.
The short answer to your question is, no, to the best of my knowledge.
There are some schemes called multi-stage secret sharing, where cheaters can be detected, to some extend. See the research announcement here and the references mentioned:
[They] introduce new rational multi-secret sharing scheme that has high security and takes an identity authentication for the dealer in distribution phase so that it is feasible to prevent the forger from cheating.
There are also schemes called verifiable secret sharing where (with very high probability) the scheme allows players to be certain that no other players are lying about the contents of their shares. This requires Multi Party Computation. Note that this attack is even simpler, when $k$ people agree to recover the share, one of them cheats and ends up learning the secret while hiding it from others (since the attacker knows her real secret).
