In cryptography, a ring signature is a type of digital signature that can be performed by any member of a set of users that each have keys. Therefore, a message signed with a ring signature is endorsed by someone in a particular set of people. One of the security properties of a ring signature is that it should be computationally infeasible to determine which of the set's members' keys was used to produce the signature. Ring signatures are similar to group signatures but differ in two key ways: first, there is no way to revoke the anonymity of an individual signature; and second, any set of users can be used as a signing set without additional setup. Ring signatures were invented by Ron Rivest, Adi Shamir, and Yael Tauman Kalai, and introduced at ASIACRYPT in 2001. The name, ring signature, comes from the ring-like structure of the signature algorithm.
Definition
Suppose that a set of entities each have public/private key pairs, (P1, S1), (P2, S2), ..., (Pn, Sn). Party i can compute a ring signature σ on a message m, on input (m, Si, P1, ..., Pn). Anyone can check the validity of a ring signature given σ, m, and the public keys involved, P1, ..., Pn. If a ring signature is properly computed, it should pass the check. On the other hand, it should be hard for anyone to create a valid ring signature on any message for any set without knowing any of the private keys for that set.
Applications and modifications
In the original paper, Rivest, Shamir, and Tauman described ring signatures as a way to leak a secret. For instance, a ring signature could be used to provide an anonymous signature from "a high-ranking White House official", without revealing which official signed the message. Ring signatures are right for this application because the anonymity of a ring signature cannot be revoked, and because the group for a ring signature can be improvised. Another application, also described in the original paper, is for deniable signatures. Here the sender and the recipient of a message form a group for the ring signature, then the signature is valid to the recipient, but anyone else will be unsure whether the recipient or the sender was the actual signer. Thus, such a signature is convincing, but cannot be transferred beyond its intended recipient. There were various works, introducing new features and based on different assumptions:
Threshold ring signatures Unlike standard "t-out-of-n" threshold signature, where t of n users should collaborate to sign a message, this variant of a ring signature requires t users to cooperate in the ring signing protocol. Namely, t parties S1, ..., St ∈ {P1, ..., Pn} can compute a (t, n)-ring signature, σ, on a message, m, on input (m, S1, ..., St, P1, ..., Pn). Linkable ring signatures The property of linkability allows one to determine whether any two signatures have been produced by the same member (under the same private key). The identity of the signer is nevertheless preserved. One of the possible applications can be an offline e-cash system. Traceable ring signature In addition to the previous scheme the public key of the signer is revealed (if they issue more than one signatures under the same private key). An e-voting system can be implemented using this protocol.
Efficiency Most of the proposed algorithms have asymptotic output size O ( n ) {\displaystyle O(n)} ; i.e., the size of the resulting signature increases linearly with the size of input (number of public keys). That means that such schemes are impracticable for real use cases with sufficiently large n {\displaystyle n} (for example, an e-voting with millions of participants). But for some application with relatively small median input size such estimate may be acceptable. CryptoNote implements O ( n ) {\displaystyle O(n)} ring signature scheme by Fujisaki and Suzuki in p2p payments to achieve sender's untraceability. More efficient algorithms have appeared recently. There are schemes with the sublinear size of the signature, as well as with constant size.
Implementation
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