Commitment Scheme

The hiding property ensures the committed value remains secret until it is revealed. The commitment string (or commitment) reveals no information about the original value. This is typically achieved using a one-way function or encryption, often with a random nonce or blinding factor to prevent brute-force attacks. For example, a Pedersen commitment uses a random blinding factor to perfectly hide the committed data.

The binding property guarantees that the committing party cannot change the value after the commitment is made. Once a commitment is published, it is computationally infeasible to find a different value that opens to the same commitment string. This prevents the committer from later claiming they committed to a different value. In some schemes, like Pedersen commitments, this property relies on the hardness of the discrete logarithm problem.

Several cryptographic primitives are used to build commitment schemes:

• Hash-based (e.g., SHA-256): Commit by publishing H(nonce || value). Simple and widely used.
• Pedersen Commitments: Use elliptic curve groups for perfect hiding and computational binding. Essential for confidential transactions.
• Polynomial Commitments (e.g., KZG): Allow committing to a polynomial and proving evaluations, forming the basis for many zk-SNARKs and data availability schemes.

Commitment schemes are a foundational component of zero-knowledge proofs (ZKPs). They allow a prover to commit to secret witness data before constructing a proof. The proof then demonstrates that the committed values satisfy certain conditions without revealing them. This enables applications like zk-SNARKs for private transactions and zk-Rollups for scalable, private computation on blockchains.

Commitment schemes are ubiquitous in blockchain protocols:

• Merkle Trees: Leaf commitments are hashes of data, with the root hash acting as a commitment to the entire set.
• Confidential Transactions: Hide transaction amounts using Pedersen commitments while preserving verifiability.
• Randomness Beacons (e.g., RANDAO): Participants commit to random values in one phase, then reveal them later to generate a collective random number.

A commitment scheme operates in two distinct phases:

1. Commit Phase: The committer sends the commitment string to the verifier.
2. Reveal Phase: The committer later sends the original value and the opening data (e.g., the nonce). The verifier runs a verification algorithm using the commitment, the revealed value, and the opening to check for consistency. This two-phase structure is crucial for protocols requiring delayed disclosure, such as fair coin tosses or sealed-bid auctions.

An additively homomorphic and information-theoretically hiding commitment scheme. It allows for the creation of a commitment to a secret value without revealing it, while enabling mathematical operations on the commitments themselves.

• Key Property: Homomorphism allows commitments to be added: Commit(a) + Commit(b) = Commit(a + b).
• Use Case: Core to confidential transactions in protocols like Mimblewimble and Monero, where transaction amounts are hidden but can still be verified.

A cryptographic data structure that commits to a set of values. It produces a single root hash that acts as a binding commitment to the entire dataset.

• Key Property: Allows efficient proofs of membership (Merkle proofs) that a specific element is contained within the committed set.
• Use Case: Ubiquitous in blockchains for committing to transaction lists (block headers) and in verifiable databases. Sparse Merkle Trees are a variant used for efficient non-membership proofs.

A scheme that commits to a polynomial, enabling efficient proofs about its evaluations. The KZG commitment, based on pairing-based cryptography, is a prominent example.

• Key Property: Allows a prover to create a constant-sized proof that a committed polynomial evaluates to a certain value at a given point.
• Use Case: Foundational for Ethereum's EIP-4844 (proto-danksharding) with data blobs and zk-SNARK proof systems, enabling scalable data availability and verification.

The simplest form, using a cryptographic hash function. A committer sends H(value, salt) as the commitment, later revealing the value and salt to open it.

• Key Property: Computationally binding and hiding, assuming the hash function is collision-resistant and pre-image resistant.
• Use Case: Found in simple cryptographic protocols, randomness beacons (commit-reveal schemes), and as a building block for more complex constructions.

Within a zk-SNARK proving system, a commitment scheme is often used to bind the prover to a witness (private data) before generating the proof. This ensures the prover cannot adapt their witness based on random challenges.

• Key Property: Integrated into the proof system's trusted setup or internal mechanics to ensure extractability and knowledge soundness.
• Use Case: Essential for the security of zk-SNARK circuits, as used in Zcash for shielded transactions and various layer-2 scaling solutions.

A cryptographic accumulator is a form of commitment to a set that allows for constant-sized membership proofs. The RSA accumulator uses a large composite modulus.

• Key Property: Supports dynamic updates (adding/removing elements) and non-membership proofs.
• Use Case: Proposed for scalable blockchain designs to replace Merkle trees for state commitments, reducing proof sizes. Used in some stateless client concepts.

A commitment's hiding property ensures the committed value remains secret before reveal. However, this security is relative to the cryptographic hash function used (e.g., SHA-256). If the input space is small (e.g., a simple 'yes/no'), an adversary can brute-force hash all possible values to break the hiding property. This necessitates commitments to values with sufficient entropy or randomness.

The binding property guarantees a committer cannot open a commitment to a different value. This depends entirely on the collision resistance of the underlying hash function. If a hash collision is found (two different inputs producing the same digest), the binding property is broken. This is a foundational risk, making the choice of a cryptographically secure hash function non-negotiable.

In blockchain contexts like commit-reveal schemes, the time delay between commitment and reveal creates a vulnerability window. A malicious actor can:

• Front-run by observing a commitment on-chain, computing its own based on that information.
• Selectively abort the reveal phase if the outcome is unfavorable, potentially stalling protocols. Mitigations include using timelocks and requiring deposits (stakes) that are slashed for non-revelation.

For schemes where the commitment (e.g., a Merkle root) represents a large dataset, security assumes the underlying data is available for verification. If the prover withholds the data after committing, the verifier cannot check the opening. This data availability problem is critical in scaling solutions and is addressed by protocols like Data Availability Sampling (DAS) and erasure coding.

Advanced commitment schemes like Polynomial Commitments (e.g., KZG) often require a trusted setup ceremony to generate public parameters. If this setup is compromised, the binding property can be broken, allowing for forged proofs. This introduces a trust assumption that must be minimized through multi-party ceremonies (e.g., Perpetual Powers of Tau) where only one participant needs to be honest.

Most practical commitment schemes rely on hash-based or discrete-logarithm-based cryptography, which is vulnerable to Shor's algorithm on a sufficiently powerful quantum computer. This threatens both binding and hiding properties in the long term. Post-quantum cryptography research focuses on developing commitment schemes based on quantum-resistant problems like lattice-based cryptography.