Are there any algorithms/protocols that let you produce a specific length of random tokens/bits in such a way that you can cryptographically determine which 2 or more tokens/bitstrings where produced using the same seed/key. Is it possible at all?

For example, if I produce these random bits: 12432, 15693, with key A and 57322, 57483 with key B. I want a third party who gets 12432 and 15693 to be able to determine they were both produced under the same key.

I thought of digital signatures as the obvious solution. However digital signatures will be to long for my requirement. I would like the token to be between 10-20 bits long, any pointers will be greatly appreciated.

Are there any algorithms/protocols that let you produce a specific length of random tokens/bits in such a way that you can cryptographically determine which 2 or more tokens/bitstrings where produced using the same seed/key. Is it possible at all?

For example, if I produce these random bits: 12432, 15693, with key A and 57322, 57483 with key B. I want a third party who gets 12432 and 15693 to be able to determine they were both produced under the same key.

I thought of digital signatures as the obvious solution. However digital signatures will be to long for my requirement. I would like the token to be between 10-20 bits long, any pointers will be greatly appreciated.

Requirements:

1. The third party should be able to verify the tokens are valid.
2. Ideally, the third party should be unable to forge tokens.
3. Tokens must be unique to a single key.

Are there any algorithms/protocols that let you produce a specific length of random tokens/bits in such a way that you can cryptographically determine which 2 or more tokens/bitstrings where produced using the same seed/key. Is it possible at all? any pointers will be greatly appreciated

For example, if I produce these random bits: 12432, 15693, with key A and 57322, 57483 with key B. I want a third party who gets 12432 and 15693 to be able to determine they were both produced under the same key.

Are there any algorithms/protocols that let you produce a specific length of random tokens/bits in such a way that you can cryptographically determine which 2 or more tokens/bitstrings where produced using the same seed/key. Is it possible at all?

For example, if I produce these random bits: 12432, 15693, with key A and 57322, 57483 with key B. I want a third party who gets 12432 and 15693 to be able to determine they were both produced under the same key.

I thought of digital signatures as the obvious solution. However digital signatures will be to long for my requirement. I would like the token to be between 10-20 bits long, any pointers will be greatly appreciated.