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Formally, anonymous credentials are two-party cryptographic protocols where a user interacts with an issuer. Such protocols were first introduced by Chaum in [Chaum], and several practical instantiations have followed.
There are two core security notions that that an anonymous credential scheme should satisfy: unforgeability an unlinkability. Informally, unforgeability protects the issues and makes sure that no unauthorized credential is spent. On the other hand, unlinkability protects the user and makes sure that the user cannot be tracked across sessions.
The literature proposes many more additional features that extend the properties of the scheme, such as embedding (authenticated) public metadata, revocation, and blacklisting. For readability, we have compressed them below.
Unforgeability refers to the notion that an adversary should not be able to create credentials without the appropriate authorizations. It should not be able to do so even after interacting multiple times with an issuer.
Definitions differ across various papers. For example, the works of [CDHK15], and [HLR15] give specific definitions that differ depending on the actual scheme.
In the case of tokens, we generally ask one-more-unforgeability, i.e. that it is not possible to spend more tokens than have been issued. In the case of credentials, we generally ask the adversary cannot authenticate for a credentials that was not issued in the first place, even after observing credentials of other users.
Unlinkability, also known as blindness, or context-hiding in other works, indicated that the redemption of a credential (where the user is anonymous) cannot be linked to its respective issuance (where the user is identified).
Definitions of unlinkability for anaonymous credential schemes appeared earliest in the work of [CL01] and subsequently in [CKLM13], [CDHK15], and [HLR15].
An anonymous credential scheme where linking the token to a user is protected by information-theoretic security guarantees has perfect, or strong unlinkability. This property has been formally studied in [AMO08], and guarantees that attacks on classical cryptosystems by unbounded adversaries will not allow linkage of activity in the future.
Anonymous tokens and credentials may introduce additional features, for helping thwi solving specific problems.
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A system is said to have concurrent security if the signer or issuer of tokens or credentials can engage in multiple, concurrent issuances at the same time without affecting security. Many schemes only include proofs that consider sequential security, wherein at most one issuance flow occurs at a time, yet fail to achieve concurrent security. In practice, concurrent security is paramount.
Tokens or credentials may be either single- or multi-show, depending on the cryptographic construction. For example, tokens derived from the output of an OPRF protocol are typically single-show because spending the same token twice forces the client to reveal the same input twice, thereby breaking unlinkability. Credentials that permit multi-show operations (such as [CL06])
Defined here by Acar et al. and implemented using an accumulator scheme with delegatable non-membership proofs. Revocation via blocklisting was also implemented by Tsang et al..
Credentials may permit attaching associated data during an issuance transaction. Attaching metadata to credentials can lead to not guarantee the desired level of anonymity. In scenarios where metadata is permitted, credentials are only unlinkable from the set of users with the same metadata (regardless of whether it is public or private).
Anonymous credentials with "Public Attributes" can include metadata (e.g. provider, country of origin). The metadata is generated and signed by the issuer as part of the "credential issuance" protocol, and are visible by the credential holder.
Certain schemes allow "selective disclosure" of attributes during token redemption which allows the user to choose which attributes to reveal and which ones to keep secret, while other schemes require disclosing all attributes.
Credentials with "Private Attributes" include metadata that is created by and visible to the issuer but not by the credential holder. This can allow services to "shadowban" or rate-limit flagged users.
Private attributes must hold privacy: it must be infiseable for the issuer to link two credentials with the same set of private attributes.
Introduced in CL06, delegatable anonymous credential system allow participants to use their credentials anonymously, as well as anonymously delegate them to other participants.
TODO
A credential scheme where there can be multiple credential issuers which can also potentially be Byzantine. (see Coconut).