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Does a system exist where it is possible to verify a signature is from a public key in a specific set of public keys, but also not know which key it was?

The idea being you know a member of a group signed the message without reveling who it was.

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Yes: ​ https://en.wikipedia.org/wiki/Ring_signature
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Depending on what you are looking for, group signatures may also be relevant.

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Group signatures and ring signatures are of course good answers but note that if you don't want traceability or linkability you can just give everybody the same key.

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