Understanding Pedersen and Kate Commitments

Cryptographic commitments play a pivotal role in ensuring the integrity and security of digital transactions and communications. Among the myriad of commitment schemes available, Pedersen and Kate commitments stand out for their unique properties and applications. This article delves into the intricacies of both, drawing insights from a previous discussion on Kate commitments Kate Polynomial Commitments.

What are Pedersen Commitments?

Pedersen commitments are renowned for their simplicity and elegance in cryptographic circles. At its core, a Pedersen commitment allows one to commit to a chosen value while keeping it hidden, with the ability to reveal the committed value later. The magic lies in its two-fold assurance: it's both hiding (the value cannot be guessed) and binding (the committer can't change the value once committed).

The Mechanism

• Select Parameters: Choose a large prime number p and a generator g of a group of order q (where q is also a large prime number).
• Choose Secret and Randomness: Select a secret value s and a random value r.
• Commit: Compute the commitment C as C = g^s * h^r mod p, where h is another generator of the group, and it's crucial that h is chosen such that no one knows the logarithm of h to the base g.
• Verification of a Pedersen Commitment
  • Reveal: The committer reveals the secret value s and the randomness r.
  • Verify: The verifier checks if C = g^s * h^r mod p. If the equation holds, the commitment is valid.

What are Kate Commitments?

Kate commitments, named after their inventor, are a more recent innovation in the field of cryptographic commitments. They are particularly adept at handling polynomial commitments, which are crucial in various cryptographic protocols, including zero-knowledge proofs.

The Mechanism

• Setup: Choose a large prime p, a generator g, and a secret point α.
• Commit to a Polynomial: Given a polynomial f(x), compute the commitment C as C = g^{f(α)} mod p.
• Verification of a Kate Commitment
  • Reveal: The committer provides a value f(x) and a proof π.
  • Verify: The verifier checks if g^{f(x)} = C * g^{-xα} * π. The proof π is typically a quotient of polynomials that can be efficiently verified.

Practical Example

Consider a voting scenario. Each voter can commit to their choice without revealing it. Later, these commitments can be opened to count votes, ensuring both privacy and integrity.

In the context of a practical example like a voting scenario, commitments ensure privacy through the commitment phase but require revealing the secret value during the verification phase. Here's how privacy is managed:

• During Voting (Commitment Phase): Each voter commits to their choice by generating a commitment without revealing their actual vote. This commitment, a cryptographic representation of their vote, is made public, but it doesn't reveal the actual vote due to the hiding property of the commitment.
• After Voting (Reveal Phase): To count the votes, each voter reveals their secret value (the actual vote) and the random value used to generate the commitment. This allows for the verification of each vote without altering it.
• Ensuring Privacy: The privacy of each vote is maintained during the voting process. Once the voting period ends, the votes are revealed for counting. The key aspect here is that no one can know or change the vote after it has been cast, maintaining the integrity of the voting process.

Can the Secret be Not Revealed?

• Pedersen Commitments: In contexts like voting, the secret (the vote) must eventually be revealed for the process to be completed (i.e., votes counted). However, there are scenarios where the actual value might never need to be revealed. For instance, in certain cryptographic protocols, the mere fact that someone possesses a commitment to a particular value is sufficient, and the actual value doesn't need to be disclosed.
• Kate Commitments: In many applications of Kate commitments, especially in zero-knowledge proofs, the actual data (secret) does not need to be revealed. Instead, what is demonstrated is the knowledge of the data satisfying certain conditions without revealing the data itself. This is particularly useful in scenarios where verifying the correctness of data is essential without exposing the data, such as in blockchain transactions.

Pedersen vs. Kate: Choosing the Right Tool

While both commitment schemes serve the purpose of secure commitments, their use cases differ significantly.

Comparison of Mathematical Operations

Creation Phase

• Pedersen: Involves exponentiation and multiplication in a finite field. It's simple yet secure due to the discrete logarithm problem, a fundamental concept in cryptography that ensures the security of the commitment.
• Kate: Focuses on committing to a polynomial, which is evaluated at a secret point. This process, especially for high-degree polynomials, can be computationally intensive, reflecting the scheme's complexity and robustness.

Verification Phase

• Pedersen: Verification is straightforward, involving checking an equation with exponentiations. This simplicity stems from the discrete logarithm problem's complexity, which ensures security.
• Kate: More intricate, as it involves not only checking an equation with exponentiations but also verifying a polynomial quotient proof. This sophisticated operation is crucial in advanced cryptographic protocols.

What to Choose

• Complexity: Kate commitments are inherently more complex due to their focus on polynomials and the proof mechanism.
• Purpose: Pedersen commitments are ideal for simple, individual commitments. In contrast, Kate commitments are designed for complex scenarios, particularly those involving polynomials. They are instrumental in advanced cryptographic protocols, notably zero-knowledge proofs.
• Privacy-centric Applications: Choose Pedersen Commitments when the focus is on individual value privacy, such as in secret voting or private auctions. Their simplicity and proven security model make them ideal for these applications.
• Simplicity and Proven Security: Opt for Pedersen Commitments in scenarios where simplicity of implementation and a well-tested security model are crucial.
• Handling Complex Data Structures: Select Kate Commitments for scenarios involving complex data structures like polynomials or multiple data points, typical in blockchain scalability solutions.
• Efficiency in Verifications: Use Kate Commitments when your application demands efficient verification of complex states or transactions, as in various cryptographic protocols including zero-knowledge proofs.

Conclusion

Pedersen and Kate commitments, each with their distinct advantages, cater to different needs in the cryptographic landscape. While Pedersen commitments offer a straightforward approach for individual commitments, Kate commitments provide an efficient solution for complex data structures. Understanding their differences and strengths is key to choosing the right tool for your cryptographic requirements.