CL Signature

A CL signature allows a signer to issue a signature on a committed message without learning the message's content. This is the core mechanism for creating anonymous credentials, where a user can obtain a credential (e.g., proof of age) without revealing their identity to the issuer.

• Process: The user sends a Pedersen Commitment of their secret data to the signer.
• Result: The signer produces a valid signature on the hidden value, which the user can later reveal and prove knowledge of.

A holder of a CL signature can cryptographically prove possession of the signature and disclose only specific attributes from the signed message set, while keeping the rest hidden. This enables fine-grained, privacy-preserving authentication.

• Example: A credential may contain {name, age, country}. The user can prove they are over 21 and from the US without revealing their exact age or name.
• Mechanism: Uses zero-knowledge proofs to demonstrate the signature is valid for the disclosed subset.

Multiple CL signatures, potentially from different issuers, can be merged into a single, compact signature. This is critical for building efficient systems where a user holds credentials from multiple sources.

• Benefit: Reduces the proof size and computational cost when demonstrating possession of several credentials simultaneously.
• Use Case: Proving you have a valid driver's license (issuer: DMV) and a professional certification (issuer: Board) in one combined proof.

CL signatures are built on the Strong RSA assumption or pairing-based cryptography, providing strong security guarantees. Their structure is defined over a cyclic group, allowing for the efficient proofs and transformations that enable their privacy features.

• Core Components: The signature is a tuple (A, e, v) where e is a large prime, allowing for the blind issuance process.
• Flexibility: The scheme can be instantiated in different cryptographic settings to optimize for proof size or verification speed.

CL signatures are the cryptographic engine for anonymous credential systems like Microsoft's U-Prove and IBM's Idemix. They solve the fundamental problem of proving qualifications without creating correlatable identifiers.

• Flow: 1) Issuance: User gets a blind signature on their attributes. 2) Presentation: User creates a zero-knowledge proof of a valid signature and discloses necessary claims.
• Property: Provides unlinkability between different presentations of the same credential.

CL signatures form a basis for advanced signature schemes. In group signatures, a member can sign on behalf of a group while remaining anonymous within it, with a designated manager able to reveal the signer's identity if needed. Variants of CL signatures are used to construct efficient ring signatures, where a signer can prove a signature came from a member of a set (a ring) without revealing which member.

CL signatures can create untraceable digital cash or tokens. A bank issues a CL-signed token representing a monetary value. The user can blindly sign this token for a merchant, who can verify the bank's signature without learning the token's unique identifier, preventing the bank from linking the withdrawal and spending transactions. This provides strong payer anonymity.

The structure of CL signatures allows them to be adapted into Verifiable Random Functions. A secret key holder can generate a pseudorandom output and a proof that it was correctly computed. This is useful in blockchain consensus mechanisms (e.g., for leader election) and proof-of-personhood systems, where a unique, non-transferable output must be generated from a credential.

CL signatures form the backbone of anonymous credential systems. They allow a user to prove they possess a valid credential (e.g., a KYC attestation or proof-of-age) issued by an authority, without revealing the credential itself or any other identifying information. This enables selective disclosure of attributes.

• Key Property: Unlinkability – multiple uses of the same credential cannot be linked together.
• Example: Proving you are over 18 from a government ID without revealing your name or birth date.

CL signatures are a succinct signature scheme that can be efficiently used within Zero-Knowledge Proofs. The signature acts as a committed value that can be proven about in a ZK circuit.

• Integration: The signature's structure allows it to be seamlessly embedded in proof systems like Groth16 or PLONK.
• Use Case: Creating a ZK proof that states, "I possess a valid CL-signed credential where attribute X=Y," without revealing the signature or other attributes.

Used in decentralized identity (DID) protocols and access control systems. CL signatures enable privacy-focused authentication where users can prove membership or authorization without a persistent on-chain identifier.

• Decentralized Identifiers (DIDs): Can sign claims associated with a DID in a privacy-preserving manner.
• Gated Access: Granting access to a dApp or service based on certified attributes (e.g., proof of unique humanity, accredited investor status) while preserving user anonymity.

Employed in ZK-Rollups and validity proofs to efficiently verify the correctness of off-chain transactions. CL signatures can aggregate or batch proofs of state transitions where user identity must remain private.

• Scalability: The small proof size of statements about CL signatures reduces on-chain verification cost.
• Privacy-Preserving Rollups: Enables transaction types where users interact based on private credentials verified by the rollup's ZK circuit.

Facilitates the creation of privacy-enhanced tokens and assets with embedded compliance. An issuer can sign tokens with attributes (e.g., region-locked, non-transferable) that holders can later prove in a private transaction.

• Compliant Assets: Enables regulatory features without sacrificing on-chain privacy for the end-user.
• Example: A security token where ownership is recorded, but trades can privately prove the seller is accredited and the buyer is in a permitted jurisdiction.

A defining feature of CL signatures is re-randomization. A user can take a valid signature and, without the secret key, create a new, unlinkable signature on the same message. This is crucial for anonymity.

• How it works: Uses mathematical operations to change the signature's randomness while preserving its validity.
• Ecosystem Impact: This property directly enables the unlinkability required for anonymous credentials and repeated, private proof presentations.

The core security of CL signatures stems from their ability to generate zero-knowledge proofs of possession. A user can prove they hold a valid signature on a set of hidden messages without revealing the signature or the messages themselves. This enables:

• Selective disclosure: Proving specific attributes from a credential.
• Unlinkability: Multiple proofs from the same credential cannot be linked together.
• Strong privacy guarantees for anonymous authentication systems.

CL signature security is formally proven under the LRSW assumption (named after Lysyanskaya, Rivest, Sahai, and Wolf). This is a computational hardness assumption in groups with a bilinear map (pairing-friendly elliptic curves). It states that, given a group generator and an oracle that produces signatures on adaptively chosen messages, it is computationally infeasible to forge a signature on a new message. This assumption is the bedrock preventing signature forgery.

A key feature is blind issuance. A user can present a blinded commitment of their messages to a signer, who produces a signature on the commitment. The user can then 'unblind' this to obtain a regular CL signature on the original messages. Crucially, the signer never learns the messages, and the process maintains existential unforgeability under adaptive chosen-message attacks (EUF-CMA), meaning an adversary cannot create a valid signature for any new message.

Theoretical security relies on correct implementation. Critical risks include:

• Randomness failures: Insecure random number generation during signature creation or proof generation can leak private keys.
• Side-channel attacks: Timing or power analysis on devices (like hardware wallets) performing the pairing operations or modular exponentiations.
• Parameter selection: Using non-standard or poorly vetted elliptic curve groups can introduce vulnerabilities. Audited libraries (like libsignal, zksnark suites) are essential.

While CL signatures provide strong anonymity, this creates tension with regulatory frameworks like Travel Rule (FATF) or KYC. Systems using them must architect privacy-enhancing compliance techniques, such as:

• Zero-knowledge KYC proofs: Proving membership in a sanctioned whitelist.
• Auditor escrow keys: Designated entities can, under legal order, decrypt transaction details.
• View keys: Users can delegate selective viewing rights. The cryptography enables compliance without default surveillance.

CL signatures are often compared and combined with other advanced primitives:

• BLS Signatures: Also use pairings, but for aggregation; CL focuses on hiding message content.
• RSA Blind Signatures: Older blind signature scheme; CL is more efficient for multiple messages and proofs.
• zk-SNARKs/STARKs: CL signatures can be used as a trusted setup for credential issuance, with the proofs executed in a zk circuit for more complex statements. Understanding these relationships is key for secure system design.

A cryptographic method where one party (the prover) can prove to another (the verifier) that a statement is true without revealing any information beyond the validity of the statement itself. CL Signatures are often used as a building block within ZKP systems to sign committed messages, enabling proofs about signed attributes without disclosing the signature or the data.

• Key Use: Enables privacy-preserving authentication and selective disclosure.
• Example: Proving you are over 21 from a signed credential without revealing your birth date.

A cryptographic primitive that allows a user to commit to a chosen value while keeping it hidden, with the ability to reveal it later. The commitment is binding (cannot change the value) and hiding (does not reveal the value). CL Signatures operate on these commitments, allowing a signer to issue a signature on a committed message without learning the message itself.

• Core Property: Binding + Hiding.
• Role in CL: Enables the 'signature on committed values' functionality.

A mathematical operation on elliptic curve groups that forms the foundation for many advanced cryptographic protocols, including CL Signatures. It takes two points from different groups and maps them to an element in a target group. This structure enables the verification equations that make CL Signatures efficient and useful for complex proofs.

• Function: Enables efficient signature verification equations.
• Protocols Used In: BLS signatures, identity-based encryption, and various ZK-SNARKs.

A digital credential system that allows users to prove they possess certain attributes (e.g., citizenship, age) without revealing their identity or other unnecessary information. CL Signatures are a core cryptographic mechanism for constructing such credentials, as they allow for the issuance of a signature on multiple attributes and subsequent unlinkable, selective disclosure of subsets.

• Key Feature: User-controlled, minimal disclosure.
• Real-World Goal: Privacy-preserving digital IDs and access tokens.

A widely used digital signature scheme based on the computational difficulty of factoring large integers. CL Signatures are a type of RSA-based signature but with a crucial enhancement: they are signatures on committed messages. This allows them to be used in protocols where the message must remain hidden from the signer, unlike standard RSA which requires the signer to know the exact message.

• Comparison: CL extends RSA for blind/committed signing.
• Underlying Math: Relies on the Strong RSA assumption.

A signature scheme where any member of a group can sign a message, producing a signature that verifies as coming from the group, but does not reveal which specific member signed it. A trusted group manager can later reveal the signer's identity if needed. While distinct, CL Signatures share conceptual goals of controlled anonymity and are sometimes used as a component in more complex group signature constructions.

• Core Property: Anonymity within a group with traceability.
• Relation: CL provides tools for attribute-based anonymity.