Homomorphic Commitment

Commit to an ordered list of values (m1, m2, ..., m_n) and later efficiently prove that m_i is the i-th element. Authenticated data structures like Merkle Trees are a simple form of vector commitment, where the homomorphic property allows for efficient updates and batch proofs.

• Advanced Schemes: RSA Accumulators and Verkle Trees offer constant-sized proofs.
• Use Case: Stateless clients in blockchain systems, where a node can verify a piece of state with a small proof.

A specialized homomorphic commitment used to prove the inner product of two committed vectors. The prover shows they know vectors a and b such that their committed inner product <a, b> equals a public value.

• Core Protocol: The Bulletproofs range proof system relies heavily on an efficient inner product argument.
• Application: Enabling compact, non-interactive zero-knowledge proofs for complex statements (e.g., proving a committed number is within a range without revealing it).

Homomorphic commitments are a critical component within zk-SNARK proving systems. The Quadratic Arithmetic Program (QAP) representing a computation is often encoded into polynomials, which are then committed using a polynomial commitment scheme.

• Process: The prover commits to polynomials representing the witness. The verifier checks evaluations of these polynomials at a random challenge point, relying on the commitment's homomorphic properties for security and efficiency.
• Result: Enables the succinct proof property central to scaling solutions.

Homomorphic commitments enable two parties to compute the intersection of their private sets without revealing the full sets to each other. One party can commit to their set, and the other can homomorphically compute on those commitments to determine common elements.

• Mechanism: Often uses additively homomorphic encryption (like Paillier) or oblivious polynomial evaluation built on commitments.
• Real-World Use: Privacy-preserving contact discovery in messaging apps, secure data matching between enterprises.

The hiding property ensures the committed value remains secret. Most practical schemes, like Pedersen commitments, offer computational hiding, which relies on the discrete logarithm problem being hard. This means an adversary with limited computational power cannot discover the value, but a theoretically unbounded adversary (e.g., a quantum computer) could. In contrast, perfectly hiding schemes (like some using Pedersen with specific parameters) provide unconditional secrecy, but then rely on computational binding.

The binding property prevents the committer from later opening the commitment to a different value. This is often based on the collision resistance of a cryptographic hash function (in hash-based commitments) or the hardness of the discrete log problem. A binding failure is a critical vulnerability, allowing fraudulent double-spends in blockchain contexts. For Pedersen commitments, binding is computational and depends on the selected elliptic curve group and generator points.

Some homomorphic commitment schemes, particularly those used in advanced zero-knowledge proofs like KZG polynomial commitments, require a trusted setup ceremony to generate public parameters. If this setup is compromised, the binding property can be broken. The security of the entire system hinges on the assumption that the toxic waste from this setup was correctly destroyed. Bulletproofs and FRI-based commitments are notable for not requiring a trusted setup.

Like all cryptography, implementations are vulnerable to side-channel attacks. These include:

• Timing attacks that analyze computation time.
• Power analysis on hardware devices.
• Fault injection to induce errors. Secure implementation requires constant-time algorithms, blinding techniques, and protection against physical tampering, especially for schemes used in wallet software or hardware security modules (HSMs).

Most widely used homomorphic commitment constructions (e.g., Pedersen, KZG) are based on elliptic curve cryptography or pairing-based cryptography, which are vulnerable to Shor's algorithm on a sufficiently powerful quantum computer. Post-quantum secure commitment schemes are an active area of research, exploring lattice-based and hash-based constructions to maintain security in a future with quantum adversaries.

Security must be evaluated within the application context:

• In confidential transactions, misuse can lead to negative or overflow amounts.
• In scalability solutions (rollups, validity proofs), a broken commitment can invalidate the entire proof system.
• Auditability is reduced, requiring reliance on cryptographic correctness rather than direct inspection, which introduces new trust assumptions for verifiers.

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. Homomorphic commitments are often a key component within ZKPs, allowing the prover to commit to secret data and then perform computations on the commitments to generate a proof.

• Example: A ZK-SNARK uses polynomial commitments, a type of homomorphic commitment, to prove correct execution of a program.

A commitment scheme that allows a prover to commit to an ordered sequence of values (a vector) with a single, short digest. It supports efficient proofs that a specific value is present at a given position. Homomorphic vector commitments enable proving properties about linear combinations of the vector's elements without revealing them.

• Use Case: Verifiable databases and stateless clients in blockchain systems, where a node can prove the state of a specific account.

A specialized form of commitment that binds a prover to a specific polynomial. It allows the prover to later reveal the polynomial's evaluation at any point and provide a proof that the evaluation is consistent with the committed polynomial. This is a homomorphic operation, as commitments to polynomials can be added to produce a commitment to their sum.

• Key Protocol: The KZG commitment scheme is a foundational polynomial commitment used in Ethereum's EIP-4844 (proto-danksharding) for data availability.

A specific, widely-used homomorphic commitment scheme based on elliptic curve cryptography. It is information-theoretically hiding and computationally binding. Its homomorphic property means that the commitment to the sum of two messages is equal to the sum of their individual commitments.

• Formula: Commit(m) = m*G + r*H, where G and H are public generator points, m is the message, and r is a blinding factor.
• Application: Foundational for confidential transactions in cryptocurrencies like Monero.

A cryptographic data structure that hashes data into a tree, producing a single root hash that commits to the entire set of data. While not inherently homomorphic for data, it is a commitment scheme used for membership proofs. Advanced constructions like a Merkle-Patricia Trie combine it with other primitives for scalable state commitments.

• Contrast: Unlike homomorphic commitments, standard Merkle trees do not easily allow proofs about computed values (e.g., the sum of leaves) without revealing the leaves.

A protocol that allows a dealer to distribute a secret among a group of participants such that only authorized subsets can reconstruct it. Participants can verify the correctness of their shares without learning the secret. Homomorphic commitments (like Pedersen commitments) are crucial to VSS, as they allow the dealer to commit to the secret polynomial and enable verifiable operations on shares.

• Application: Secure distributed key generation (DKG) for threshold cryptography in validator networks.