Pedersen Vector Proofs: An Efficient Data Commitment Scheme with Succinct Proofs

Disclaimer

This article presents theoretical research on Pedersen Vector Proofs and is intended for educational and research purposes only. The scheme described has not undergone formal security analysis or peer review. Implementation in production systems should be approached with caution and appropriate expertise.

Abstract

In this research article, we introduce the underlying concepts of Pedersen commitments and vector commitments before presenting a Pedersen Vector Proof (PVP) scheme, an efficient data commitment with succinct proofs which allows for both single-point and batch proofs. We go into the details of the favorable properties of Pedersen commitments, then explore proof generation, proof verification and provide a low-level language-independent pseudocode. We will also discuss briefly on the practical use cases of Pedersen Vector Proofs and trade-offs when compared to traditional Merkle trees.

Introduction

Commitment schemes are fundamental cryptographic primitives that enable a party to commit to a value, while being able to keep it abstracted, yet later reveal it in a verifiable manner. This property of commitment schemes makes them highly favorable for a large number of applications in cryptographic applications. Traditional Merkle trees offer efficient data integrity, but follow logarithmic proof sizes and sequential verification costs. In contrast, Pedersen commitments, known for their homomorphic properties, serve as natural building blocks for more schemes such as vector commitments. This article aims to introduce a Pedersen-based vector commitment scheme with efficient (and potentially compressible) proofs, which would be referred to here as Pedersen Vector Proof (PVP) scheme. This paper does not explicitly guarantee its practical usability, but at least somewhat of a deep technical understanding of the idea behind PVPs and how they could fit as a useful addition to already available commitment schemes.

Pedersen Commitments

Definition and Properties

A Pedersen commitment is defined over a cyclic group $G$ of prime order $q$, with two public generators $g$ and $h$. For such design, we state that for a value $x \in \mathbb{Z}_q$ and a randomly chosen blinding factor $r \in \mathbb{Z}_q$, the commitment $C$ is given by:

$$C = x \cdot g + r \cdot h$$

This arrangement provides two particular properties which makes Pedersen commitments favorable, Hiding and Binding. The randomness of $r$ makes $C$ statistically independent of $x$, which means that a verifier should (and would) not be able to derive the value of $x$ without proper knowledge of $r$. Pedersen commitments are also based on the assumption of the discrete logarithm problem, this makes it computational infeasible to open $C$ to a different value (for now, we purposely ignore any tradeoffs associated with such assumptions).

Vector commitments

Extending to Vectors

A vector commitment generalizes the concept of a commitment from a single value to an entire vector of values $\mathbf{x} = (x_1, x_2, \dots, x_n)$. By employing multiple independent generators $g_1, g_2, \dots, g_n$ (which could be, for example, derived from a hash function) and a common blinding generator $h$, we create a "vector commitment based on Pedersen commitments", we compute such vector commitment as:

$$C = \sum_{i=1}^{n} x_i \cdot g_i + r \cdot h$$

This arrangement allows us to generate a single proof $C$ for a group of elements regardless of the vector's length. Binding and Hiding properties are also inherited from Pedersen commitments.

The Pedersen Vector Proof (PVP) Commitment Scheme

Overview

Pedersen Vector Proofs leverage the vector commitment approach to allow efficient proof generation and verification for both single and multiple (batch) openings. The core idea here is to commit to a vector of data values and later reveal selective entries with proofs that are both succinct and potentially amenable to zero-knowledge extensions.

Commitment Construction

For our design, given:

• A vector $\mathbf{x} = (x_1, x_2, \dots, x_n)$
• Independent generators $g_1, g_2, \dots, g_n$
• A blinding factor $r$ and a blinding generator $h$

The commitment would be calculated as:

$$C = \sum_{i=1}^{n} x_i \cdot g_i + r \cdot h$$

where:

• $\mathbf{x} = (x_1, x_2, \dots, x_n)$ is the message vector containing the intended values.
• $\mathbf{g} = (g_1, g_2, \dots, g_n)$ is the generator vector
• $r$ is the blinding factor (random scalar).
• $h$ is an independent blinding generator.
• All elements belong to a prime-order group $\mathbb{G}$, where operations are performed modulo a prime $p$.

Proof Generation

Single Point Opening Proof

To prove that a particular value $x_j$ (where $x_j$ is a part of the message vectors $x$) is committed in $C$, the prover provides:

• The index $j$, which would indicate the location of $x_j$
• The value $x_j$
• The common blinding factor $r$

The proof $\pi$ is then:

$$\pi = (j, x_j, r)$$

Verification Equation

The verifier is able to confirm inclusion by computing:

$$C' = C - x_j \cdot g_j$$

and then further checks whether:

$$C' \stackrel{?}{=} r \cdot h$$

If equality holds to be true, the proof is accepted and inclusion of $x_j$ is proven.

Batch Proof For Multi-Point Openings

When multiple indices ${j_1, j_2, \dots, j_m}$ need to be opened simultaneously, a batch proof can be used to compress these individual proofs.

Batch Proof Generation

To compute Batch Proofs, we:

1. Select Random Scalars: For each index $j_k$ in the batch, generate a random coefficient $\beta_k$, we could achieve this by using a Fiat-Shamir heuristic.

2. We move on to compute an aggregated value, such that: $$A = \sum_{k=1}^{m} \beta_k \cdot x_{j_k} \cdot g_{j_k}$$

3. We then form the Batch Proof which would ideally consist of: $$\pi_{\text{batch}} = \left( {j_1, j_2, \dots, j_m}, {x_{j_1}, x_{j_2}, \dots, x_{j_m}}, r \right)$$ with the aggregation implicitly used in the verification step.

Batch Verification:

To verify batch proofs, the verifier uses the same random coefficients $\beta_k$ and checks: $$C - \left(\sum_{k=1}^{m} \beta_k \cdot x_{j_k} \cdot g_{j_k}\right) \stackrel{?}{=} r \cdot h$$

Based on my theory, this single aggregated check is far more efficient than verifying each individual proof separately (as seen in traditional Merkle proofs).

Summary of Proof Verification

Single Proof Verification

Given that:

• Commitment $C$
• Proof $\pi = (j, x_j, r)$
• Generators $g_1, g_2, \dots, g_n$ and $h$

We can correctly verify our given proof by:

1. Computing: $$C' = C - x_j \cdot g_j$$
2. Accepting the proof if: $$C' = r \cdot h$$

Batch Proof Verification

For a batch proof $\pi_{\text{batch}}$, we take a different means of verification, to avoid individual verification of its constituents:

1. For each opened index $j_k$, use the same random scalar $\beta_k$ (recomputed deterministically).
2. Compute the aggregated value: $$A = \sum_{k=1}^{m} \beta_k \cdot x_{j_k} \cdot g_{j_k}$$
3. Verify that: $$C - A = r \cdot h$$

Security Assumptions and Tradeoffs

The security of Pedersen Vector Proofs relies on the following key assumptions:

1. Discrete Logarithm Assumption: The binding property depends on the computational hardness of the discrete logarithm problem in the underlying group. This makes the scheme potentially vulnerable to quantum attacks using Shor's algorithm.

2. Generator Independence: The security analysis assumes the generators $g_1, g_2, \dots, g_n$ are independent and properly generated. In practice, deriving these generators securely is critical.

3. Random Number Generation: The security of the hiding property relies on truly random blinding factors. Weak random number generators could compromise confidentiality.

Practical tradeoffs include:

• Computation vs Storage: PVPs offer constant-size commitments at the cost of higher computational requirements for large vectors.
• Quantum Resistance: Unlike hash-based schemes, PVPs are not post-quantum secure.
• Efficiency vs Flexibility: While batch proofs provide efficiency, they come at the cost of increased complexity in implementation.

A Pseudocode Implementation:

Given the design, we can describe the language-independent implementation of this scheme below:

// --- Initialization ---
// Input: Number of vector elements n
// Output: Set of generators {g1, g2, ..., gn} and a blinding generator h

FUNCTION Setup(n):
    generators = []
    FOR i = 1 TO n:
        generators.append(RandomGroupElement())
    h = RandomGroupElement()
    RETURN (generators, h)

// --- Commitment Generation ---
// Input: Vector x = (x1, x2, ..., xn), blinding factor r, generators, and h
// Output: Commitment C

FUNCTION Commit(x, r, generators, h):
    C = r * h
    FOR i = 1 TO n:
        C = C + x[i] * generators[i]
    RETURN C

// --- Single Point Proof Generation ---
// Input: Index j, vector x, blinding factor r
// Output: Proof π = (j, x_j, r)

FUNCTION GenerateProofSingle(j, x, r):
    RETURN (j, x[j], r)

// --- Single Point Proof Verification ---
// Input: Commitment C, proof (j, x_j, r), generators, and h
// Output: Boolean (valid or invalid)

FUNCTION VerifyProofSingle(C, j, x_j, r, generators, h):
    C_prime = C - x_j * generators[j]
    IF C_prime == r * h THEN
        RETURN TRUE
    ELSE
        RETURN FALSE

// --- Batch Proof Generation ---
// Input: Set of indices I = {j1, j2, ..., jm}, vector x, blinding factor r, and generators
// Output: Batch proof π_batch = (I, {x_j for each j in I}, r)

FUNCTION GenerateBatchProof(I, x, r):
    proofIndices = I
    proofValues = []
    FOR each j in I:
        proofValues.append(x[j])
    RETURN (proofIndices, proofValues, r)

// --- Batch Proof Verification ---
// Input: Commitment C, batch proof (I, proofValues, r), generators, h, and function RandomCoefficient(index)
// Output: Boolean (valid or invalid)

FUNCTION VerifyBatchProof(C, I, proofValues, r, generators, h):
    A = IdentityElement() // group identity
    FOR k = 1 TO length(I):
        beta = RandomCoefficient(I[k]) // Deterministically computed via Fiat-Shamir
        A = A + beta * proofValues[k] * generators[I[k]]
    IF (C - A) == r * h THEN
        RETURN TRUE
    ELSE
        RETURN FALSE

This could serve as a blueprint for practical implementations in various programming environments and further optimizations.

Comparison Of Pedersen Vector Proof Against Other Designs

Feature	Merkle Trees	KZG Commitments	Bulletproofs	Pedersen Vector Proof (PVP)
Commitment Size	1 hash (constant)	1 group element (constant)	1 group element (constant)	1 group element (constant)
Proof Size	$O(\log n)$	$O(1)$	$O(\log n)$	Single proof: $O(1)$; Batch: $O(\log \log n)$ (compressed)
Verification Complexity	$O(\log n)$ hash ops	Constant (pairing check)	$O(\log n)$	Constant-time (or aggregated constant for batch)
Trusted Setup	None	Yes	None	None
Security Assumption	Collision resistance of hash	Discrete log in pairing groups	Discrete log	Discrete log
Zero-Knowledge Friendliness	Low (needs extra layers)	Moderate (zk-SNARKs integration)	High	High
Batch Proof Capability	Difficult to aggregate	Native, but pairing heavy	Aggregatable, $O(\log n)$	Efficient with inner product arguments
Dynamic Updates	Efficient (only path updated)	Moderate	Moderate	Less efficient (commitment recomputation)

In summary, Merkle trees are much simpler and update-friendly but require O(logn) proof sizes and verification, while KZG commitments do offer constant-size proofs and rapid pairing-based verification, this comes at the cost of a trusted setup and practically higher computational overhead. Bulletproofs provides zero-knowledge, aggregatable proofs with logarithmic proof sizes, though generating them could be computationally intensive. In contrast, the Pedersen Vector Proof (PVP) scheme delivers compact, constant-size commitments with efficient, batchable verification and natural zero-knowledge integration, though it is less efficient for dynamic updates and relies on discrete log assumptions, making it less quantum-resistant.

Practical Use Cases

Blockchain Rollups and Data Availability: The PVP scheme commits large data vectors into a single group element, which could potentially reduce on-chain storage. The advantage of constant-time (or near-constant in the case of batch proofs) presents potential possibilities for scalability and low gas costs.

Zero-Knowledge Proof Systems: Pedersen Vector Proofs are designed with compatibility with inner product arguments, which enables further proof compression and also non-interactive verification, potentially allowing integration with zk-SNARKS/Bulletproofs. The blinding factor $r$ guarantees that the committed data remains hidden until explicitly opened.

Secure Data Aggregation: Pedersen Vector Proofs have the potential to be highly useful in scenarios where multiple data points need to be proven simultaneously and efficiently, a standard example is aggregated sensor data in IoT devices or in Multi-attribute credentials.

Conclusion

This research article has presented a mathematical and algorithmic overview of Pedersen Vector Proofs Commitment Scheme. By extending the traditional Pedersen commitments to vectors and providing native support for batch proofs, the design offers a potential lightweight alternative to Merkle trees with constant-sized commitments and efficient verifications. While this article is solely intended to act as a well-structured overview of the scheme, I intentionally omitted formal security proofs and discussed very little of practical use cases, this would be properly discussed in future articles as I continue to research PVPs.