Like centralised credentials, Nym's Coconut credentials can be signed with a secret key and later verified by anybody with the corresponding public key. But Nym credentials are much more powerful than “normal” signature schemes like standard RSA or DSA.
Specifically, Coconut is a blinded, re-randomizable, selective disclosure threshold credential signature scheme. That's quite a mouthful, so let's break it all down into digestible chunks.
Let's say you have a message in hand. In addition to the normal
sign(message, secretKey) and
verify(message, publicKey) functions present in other signature schemes, Coconut adds the following:
Blind signatures - disguises message content so that the signer can't see what they're signing. This defends users against signers: the entity that signed can't identify the user who created a given credential, since they've never seen the message they're signing before it's been blinded (turned into gobbledygook). Coconut uses zero-knowledge proofs so that the signer can sign confidently without seeing the unblinded content of the message.
Re-randomizable signatures - take a signature, and generate a brand new signature that is valid for the same underlying message. The new bitstring in the re-randomized signature is equivalent to the original signature but not linkable to it. So a user can “show” a credential multiple times, and each time it appears to be a new credential, which is unlinkable to any previous “show”. But the underlying content of the re-randomized credential is the same (including for things like double-spend protection). This once again protects the user against the signer, because the signer can't trace the signed message that they gave back to the user when it is presented. It also protects the user against the relying party that accepts the signed credential. The user can show re-randomized credentials repeatedly, and although the underlying message is the same in all cases, there's no way of tracking them by watching the user present the same credential multiple times.
Selective disclosure of attributes - allows someone with the public key to verify some, but not all, parts of a message. So you could for instance selectively reveal parts of a signed message to some people, but not to others. This is a very powerful property of Coconut, potentially leading to diverse applications: anonymous voting systems, selective revelation of medical data, etc.
Threshold issuance - allows signature generation to be split up across multiple nodes and decentralised, so that either all signers need to sign (n of n where n is the number of signers) or only a threshold number of signers need to sign a message (t of n where t is the threshold value).
Taken together, these properties provide excellent privacy for Nym users when it comes to generating and using signatures for cryptographic claims.