Efficient Vector Commitments with Sublinear Update Information and Proof Updates

Core Concepts

The authors propose a family of vector commitment schemes that can achieve both sublinear update information size and sublinear runtime complexity for proof updates, by leveraging homomorphic properties of the underlying tree-based data structures.

Abstract

The paper introduces the concept of homomorphic trees, where the inner nodes of the tree can be expressed as homomorphic functions of the underlying committed messages. This allows for efficient updates to the tree structure when the committed messages are changed, without requiring the users to keep track of the entire tree or recalculate it from scratch.
The key insights are:

Homomorphic trees enable calculating the new inner node values after message updates using only the new and old message values, or their differences. This is in contrast to Merkle trees, where updating the tree requires publishing the changes to all affected inner nodes.

The authors propose a family of homomorphic tree constructions, parameterized by a trade-off parameter ν ∈ (0, 1). These constructions achieve both sublinear update information size (Θ(kν)) and sublinear runtime complexity for proof updates (Θ(k1-ν)), where k is the number of updated messages.

The authors prove an information-theoretic lower bound on the relation between the update information size and the proof update runtime, which shows the asymptotic optimality of their constructions.

The authors discuss applications of their vector commitment schemes, including supporting stateless clients in Ethereum and databases with frequent membership changes.

Overall, the paper presents an efficient solution to the challenge of designing vector commitments that minimize both the update information size and the proof update complexity, which is crucial for various applications.

Stats

The size of the update information |U| is Θ(kν) and the runtime complexity of proof updates is Θ(k1-ν), where k is the number of updated messages and ν ∈ (0, 1) is a trade-off parameter.

Quotes

"We propose a family of VCs that can support sublinear update, where both the length |U| of the update information and the complexity of proof updates are sublinear in k."
"Our candidate construction with sublinear update is a homomorphic tree, based on the VCs from [45,40], where each inner node can be expressed as a sum of the partial digests of the messages underneath the node and its sibling."

Key Insights Distilled From

Vector Commitments with Efficient Updates

by Ertem Nusret... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2307.04085.pdf

Deeper Inquiries

How can the homomorphic tree constructions be made secure against quantum computers, without requiring a trusted setup and large public parameters

To make homomorphic tree constructions secure against quantum computers without the need for a trusted setup and large public parameters, one approach is to utilize post-quantum cryptographic primitives. Specifically, replacing the underlying cryptographic assumptions with quantum-resistant alternatives can enhance the security of the homomorphic trees. For example, instead of relying on traditional cryptographic hash functions vulnerable to quantum attacks, adopting quantum-resistant hash functions like SHA-3 or cryptographic schemes based on lattice-based cryptography can bolster the security of the homomorphic trees.
Additionally, implementing quantum-resistant digital signature schemes, such as those based on hash-based signatures or code-based cryptography, can further fortify the security of the homomorphic tree constructions. By integrating these post-quantum cryptographic techniques into the design and implementation of the homomorphic trees, it is possible to mitigate the threats posed by quantum adversaries while maintaining the integrity and confidentiality of the data stored within the structures.

Can the homomorphic properties of the tree-based data structures be leveraged to achieve additional functionality, such as efficient batch updates or aggregation of opening proofs

The homomorphic properties of tree-based data structures, such as homomorphic trees, can indeed be leveraged to achieve additional functionality beyond traditional vector commitments. One key advantage is the ability to efficiently perform batch updates on the committed data. By exploiting the homomorphic nature of the tree structure, multiple updates to the underlying messages can be aggregated and processed simultaneously, reducing the overall computational overhead and improving the efficiency of updating the commitments.
Furthermore, the homomorphic properties of the tree-based data structures enable the aggregation of opening proofs. This aggregation capability allows for the succinct representation of multiple opening proofs as a single proof, streamlining the verification process and reducing the computational complexity associated with verifying individual proofs. By consolidating multiple proofs into a single aggregated proof, the efficiency and scalability of the verification process can be significantly enhanced, making the system more practical and resource-efficient.

What are the practical considerations and trade-offs involved in choosing the optimal value of the trade-off parameter ν for different real-world applications of vector commitments

When selecting the optimal value of the trade-off parameter ν for different real-world applications of vector commitments, several practical considerations and trade-offs need to be taken into account. The choice of ν impacts both the size of the update information and the complexity of proof updates, influencing the overall performance and efficiency of the vector commitment scheme.
For applications where minimizing the size of the update information is crucial, a lower value of ν may be preferred to achieve sublinear update information size. This is particularly important in scenarios where bandwidth or storage constraints are significant factors, such as in blockchain systems or distributed databases.
On the other hand, if the focus is on reducing the computational complexity of proof updates, a higher value of ν may be more suitable. This can be advantageous in applications where the computational resources of the users are limited, and optimizing the runtime efficiency of proof updates is paramount.
Ultimately, the optimal value of ν will depend on the specific requirements and constraints of the application, balancing the need for efficient proof updates with the desire to minimize the size of the update information. Conducting thorough performance evaluations and considering the unique characteristics of the application environment will help determine the most suitable value of ν for achieving the desired balance between update information size and proof update complexity.