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I understand that Shamir's Secret sharing has nothing to do with the methods that are used to share and later recover a key.

That means then, the secret is only as secure as the methods used in sharing the key, because I can try to impersonate the dealer.

If the dealer shares their private key in some fashion, but later loses the private key, the dealer needs to contact enough shareholders (k out of N) to reconstruct the key.

If I assume this is done in a naive fashion, e.g. a web request, then anybody can do that.

If we secure the way is done, let's say with a password, then the key is only as secure as the password.

If the list of shareholders is secret, then the it's only as secure as the list of shareholders (where and how it is stored).

If we secure the communications with a private/public key method, then the dealer needs to have a second private key that they can use when they've lost the first one, and the key is only as secure as the second private key. The dealer is trying to secure the first private key by sharing... but then gets a second private key they need to secure. That doesn't make sense.

Is there a way to solve this problem? Or - this is my conclusion - is it that secret sharing is simply not suitable for keeping a key secure?

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Yeah, as stated, that's an unsolvable problem.

Basically, if you assume that the dealer loses everything that would let the shareholders (or anyone else) distinguish them from some random stranger, then there's nothing they can do that some random stranger couldn't also do.

The typical real world solution is to let the shareholders identify you by some physical means, such as by your fingerprints, iris patterns, DNA or just general familiarity with your physical appearance and personality. Such biometric IDs are hard to accidentally lose (especially if there's some redundancy) and even harder for someone else to copy (at least in a way that would fool a careful human inspector; unfortunately automatic checks are often quite a bit easier to trick).

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