Triptych

Triptych was designed to address limitations in Monero's previous RingCT system. Its key improvements include:

• Logarithmic Scaling: Signature size grows with O(log n) of the ring size, not O(n). This enables larger anonymity sets without a linear increase in blockchain bloat.
• Reduced Verification Cost: More efficient batch verification reduces the computational load on nodes.
• Flexibility: Supports non-power-of-two ring sizes, providing more configuration options for the protocol.

The protocol provides specific cryptographic guarantees that define its privacy properties:

• Anonymity: The true signer is hidden within a set of decoy outputs (the ring). An external observer cannot determine the actual spent output with probability better than random guessing.
• Linkability: It prevents double-spending; if the same key image is used twice, the two transactions can be linked as fraudulent.
• Unforgeability: Only a holder of a valid secret key can produce a signature that passes verification.

For a blockchain to use Triptych, it must be integrated at the consensus protocol level. This involves:

• Transaction Validation: Full nodes must verify Triptych proofs for every transaction.
• Wallet Construction: Wallets must be able to generate proofs, select decoy outputs from the blockchain's UTXO set, and create valid signatures.
• Network Rules: The protocol's parameters (like minimum ring size) are enforced by all network participants.

Understanding Triptych requires familiarity with underlying cryptographic primitives and related privacy solutions:

• Ring Signatures: The foundation for hiding the sender among a group.
• Confidential Transactions (CT): Hides transaction amounts using Pedersen commitments.
• Zero-Knowledge Proofs: Allows proof of knowledge without revealing the secret.
• Mimblewimble: An alternative privacy approach using aggregation and cut-through.
• zk-SNARKs: A different class of zero-knowledge proofs used in protocols like Zcash.

Triptych is built upon the LSAG signature scheme, a core cryptographic primitive for Monero's privacy. It allows a signer to prove membership in a group (an anonymity set) without revealing which specific member they are. The 'linkable' property ensures a signer cannot double-spend the same funds, as two signatures from the same key can be publicly linked and rejected.

• Anonymity Set: The set of possible signers in a transaction, which in Triptych can be exponentially larger than in previous schemes.
• Spontaneous: The group can be formed ad-hoc without requiring pre-existing setup or coordination among members.

A key innovation of Triptych is that the size of its zero-knowledge proofs grows logarithmically with the size of the anonymity set, unlike earlier linear schemes. This means:

• Scalability: Doubling the ring size (e.g., from 64 to 128 members) adds only a constant amount of data to the proof, not double the data.
• Efficiency: Enables much larger practical ring sizes (e.g., 2^16 members) without a corresponding explosion in transaction size, significantly strengthening privacy.

The protocol's core is a one-out-of-many proof, a type of zero-knowledge proof. It allows a prover to convince a verifier that they know a secret corresponding to one of many public keys (the ring), without revealing which one. Triptych implements this using sophisticated commitments and polynomial evaluations to achieve its logarithmic efficiency.

• Zero-Knowledge: Reveals nothing beyond the validity of the statement.
• Soundness: It is computationally infeasible to create a valid proof for a false statement.

Triptych's security relies on well-established cryptographic assumptions, similar to other privacy coins. The primary assumptions are:

• Discrete Logarithm Problem (DLP): The computational hardness of solving discrete logarithms in a cryptographic group.
• Decisional Diffie-Hellman (DDH): The difficulty of distinguishing between specific tuples of group elements.

These are standard assumptions in elliptic curve cryptography, providing confidence in the protocol's foundational security against forgery and anonymity breaking.

Triptych manages a critical trade-off inherent to private transactions: unlinkability for privacy versus linkability to prevent double-spending.

• Unlinkability: Transactions cannot be linked to each other or to a specific user, providing strong anonymity.
• Linkability: Only the specific case of a double-spend attempt creates a linkable signature, which is publicly verifiable and results in the rejection of the fraudulent transaction. This targeted linkability preserves privacy for all honest users.

As with any cryptographic protocol, real-world security depends on correct implementation and rigorous review.

• Side-Channel Attacks: Implementations must be constant-time to avoid leaking secret key material through timing or power analysis.
• Protocol Upgrades: Integrating Triptych into a live blockchain like Monero requires a hard fork and careful consensus management.
• Third-Party Audits: The novel cryptographic constructs require extensive peer review and formal security audits to verify the theoretical security proofs hold in practice.

A privacy-enhancing signature scheme where a signer can anonymously prove membership in a group. In Triptych, ring signatures are a core component, allowing a prover to demonstrate they own one of many possible spent outputs without revealing which one.

• Key Concept: Anonymity Set – the size of the group determines the privacy level.
• Comparison: Unlike basic ring signatures, Triptych's construction is more efficient, enabling much larger anonymity sets without a linear increase in proof size.

Cryptographic protocols that allow one party (the prover) to convince another (the verifier) that a statement is true without revealing any information beyond the validity of the statement itself. Triptych utilizes a specific form of non-interactive zero-knowledge proof to validate transactions.

• Role in Triptych: Proves that a new transaction output is correctly formed, the inputs are legitimate and unspent, and the signer owns a valid secret key—all without linking inputs to outputs or revealing amounts.

A cryptographic algorithm that allows a user to commit to a chosen value while keeping it hidden, with the ability to later reveal it. The commitment is binding (cannot be changed) and hiding (does not reveal the value).

• Use in Triptych: Pedersen Commitments are used to hide transaction amounts. The protocol ensures that the sum of input commitments equals the sum of output commitments plus a commitment to the transaction fee, proving no new money was created without revealing the amounts.

An extension of ring signatures that adds a tagging mechanism. While providing signer anonymity within a ring, it allows anyone to detect if the same signer created two different signatures, preventing double-spending.

• Contrast with Triptych: Triptych is a non-linkable ring signature. It prevents tracing of individual transactions but does not include a linkability tag by default, as double-spending protection is handled by other consensus mechanisms (like checking if an output has been spent).

A method for generating a unique, one-time public address for each transaction sent to a recipient, breaking the on-chain link between the recipient's public view key and their received funds.

• Relation to Triptych: While Triptych focuses on hiding the sender's identity via ring signatures, one-time addresses are often used in conjunction to protect the receiver's identity, creating a comprehensive privacy system for both sides of a transaction.

Short, non-interactive zero-knowledge proofs used primarily for proving statements about committed values, such as "this committed amount is within a valid range." They are succinct, meaning their size grows logarithmically with the statement complexity.

• Connection to Triptych: Triptych's efficiency gains are similar in spirit to Bulletproofs. While Triptych optimizes ring signature proofs, Bulletproofs++ is a later development that further reduces the size of range proofs, which are a critical component for confidential transactions in privacy protocols like Monero.