A Group signature scheme is a method for allowing a member of a group to anonymously sign

Digital signature

A digital signature or digital signature scheme is a mathematical scheme for demonstrating the authenticity of a digital message or document. A valid digital signature gives a recipient reason to believe that the message was created by a known sender, and that it was not altered in transit...

 a message on behalf of the group. The concept was first introduced by David Chaum

David Chaum

David Chaum is the inventor of many cryptographic protocols, including blind signature schemes, commitment schemes, and digital cash. In 1982, Chaum founded the International Association for Cryptologic Research , which currently organizes academic conferences in cryptography research...

 and Eugene van Heyst in 1991. For example, a group signature scheme could be used by an employee of a large company where it is sufficient for a verifier to know a message was signed by an employee, but not which particular employee signed it. Another application is for keycard access to restricted areas where it is inappropriate to track individual employee's movements, but necessary to secure areas to only employees in the group.

Essential to a group signature scheme is a group manager, who is in charge of adding group members and has the ability to reveal the original signer in the event of disputes. In some systems the responsibilities of adding members and revoking signature anonymity are separated and given to a membership manager and revocation manager respectively. Many schemes have been proposed, however all should follow these basic requirements:

• Soundness and Completeness: Valid signatures by group members always verify correctly, and invalid signatures always fail verification.
• Unforgeable: Only members of the group can create valid group signatures.
• Anonymity: Given a message and its signature, the identity of the individual signer cannot be determined without the group manager's secret key.
• Traceability: Given any valid signature, the group manager should be able to trace which user issued the signature. (This and the previous requirement imply that only the group manager can break users' anonymity.)
• Unlinkability: Given two messages and their signatures, we cannot tell if the signatures were from the same signer or not.
• No Framing: Even if all other group members (and the managers) collude
  Collusion
  Collusion is an agreement between two or more persons, sometimes illegal and therefore secretive, to limit open competition by deceiving, misleading, or defrauding others of their legal rights, or to obtain an objective forbidden by law typically by defrauding or gaining an unfair advantage...
  
  , they cannot forge a signature for a non-participating group member.
• Unforgeable tracing verification: The revocation manager cannot falsely accuse a signer of creating a signature he did not create.

The ACJT 2000, BBS04 (in Crypto), BS04 (in CCS) group signature schemes are the state of the art. (Note: this might be an incomplete list)

BBS04: Boneh, Boyen and Shacham published in 2004 (Crypto04) a novel group signature scheme based on bilinear maps. Signatures in this scheme are approximately the size of a standard RSA signature (around 200 bytes). The security of the scheme is proven in the random oracle model and relies on the Strong Diffie Hellman assumption (SDH) and a new assumption in bilinear groups called the Decision linear assumption (DLin).

A more formal definition that is geared towards provable security

Provable security

In cryptography, a system has provable security if its security requirements can be stated formally in an adversarial model, as opposed to heuristically, with clear assumptions that the adversary has access to the system as well as enough computational resources...

 was given by Bellare, Micciancio and Warinschi.

See also

• Ring signature
  Ring signature
  In cryptography, a ring signature is a type of digital signature that can be performed by any member of a group of users that each have keys. Therefore, a message signed with a ring signature is endorsed by someone in a particular group of people...
  
  : A similar system that excludes the requirement of a group manager and provides true anonymity for signers.

• Threshold signature: A threshold signature involves a fixed-size quorum (threshold) of signers. Each signer must be a genuine group member with a share of a group secret signing key. A (t,n) threshold signature scheme supports n potential signers, any t of which can on behalf of the group. Threshold signatures reveal nothing about the t signers; no one can trace the identity of the signers (not even a trusted center who have set up the system).

• Multisignature: A multisignature represents a certain number of signers signing a given message. Number of signers is not fixed and signers identities are evident from a given multi-signature. A multisignature is much shorter (sometimes constant) than the simple collection of individual signatures.

• Proxy signature: A proxy signature allows a delegator to give partial signing rights to other parties called proxy signers. Proxy signatures do not offer Anonymity

• Identity Escrow Schemes: Interactive dual of group signatures. Instead of off-line generation, a signature is directly generated by a signer based on a challenge provided by the verifier.