A Deep Dive into SNARKs: Polynomial Commitments, Merkle Trees, and Efficient Proofs
Structure:
Introduction:
In the rapidly evolving realm of cryptography, Succinct Non-interactive Arguments of Knowledge (SNARKs) stand out as powerful tools for designing efficient and secure protocols. SNARKs are proofs that allow a prover to convince a verifier that a certain statement is true, without revealing any extra information and requiring only a small amount of computation from the verifier. This article provides a comprehensive exploration of SNARKs, highlighting how they can be generated from interactive proofs, the concept of polynomial and multilinear polynomial commitments, the application of Merkle trees in creating these commitments, and the challenges involved in transforming these constructs. We delve into a detailed discussion of the foundational elements of these cryptographic proofs, their applications, and potential improvements in their efficiency. Whether you're a seasoned cryptographer or new to the field, this article offers valuable insights into the intricacies of SNARKs.
Recall: The trivial SNARK is not a SNARK
To recall the definition of a SNARK and why a trivial SNARK is not actually considered a SNARK, we need to discuss this in the context of SNARKs for the circuit satisfiability problem. Here, the prover claims to know a witness 'w' that causes a specific circuit 'C' to output zero. The trivial proof system for this problem would be for the verifier to simply receive the witness 'w' from the prover and then check that it indeed causes the circuit to output zero. However, this approach is not considered succinct due to two reasons: first, the witness 'w' could be too lengthy, making the proof longer than necessary, and second, evaluating the circuit with input 'x' followed by 'w' could be too complex, requiring the verifier to be faster than the existing system. Therefore, to speed up the verifier, we use an interactive proof after the prover sends the witness to the verifier.
SNARKS from Interactive Proofs (IPs)
To generate SNARKs from Interactive Proofs, we apply an interactive proof that allows the prover to confirm that 'w' satisfies the claimed property, i.e., evaluating the circuit 'C' on input 'x' followed by 'w' equals zero. This situation is similar to the example I introduced while discussing interactive proofs, where the verifier uses the interactive proof to offload a challenging computation such as running an expensive computer program or evaluating a specific circuit on a public input like 'w' to an untrusted prover. Here, when the prover sends 'w' to the verifier, 'w' becomes public and known to both parties. We can use an interactive proof to compel the prover to do the hard work of evaluating the circuit on 'x' followed by 'w', and ultimately prove to the verifier that this computation is being carried out correctly. However, this process still requires the prover to explicitly send 'w' to the verifier, making the proof too lengthy.
To address this, the prover, instead of explicitly sending the witness to the verifier, succinctly commits to the witness using a cryptographic commitment scheme. Now, as the prover has not sent 'w' explicitly to the verifier, it has only cryptographically committed to 'w'. This necessitates the use of a cryptographic commitment scheme that allows the prover to reveal enough information about the committed witness for the verifier to perform the necessary checks in the interactive proof protocol. This commitment and revealing of information helps to emulate random challenges from the verifier, despite the verifier not actually selecting them.
Recall: three important functional commitments
Three important types of functional commitments are Polynomial Commitments, Multilinear Polynomial Commitments, and Vector Commitments.
In a Polynomial Commitment, the untrusted prover is able to commit to a univariate polynomial 'F' of degree at most 'd'. Later, if the verifier requests to know the evaluation of that polynomial at a specific point 'r', the prover can reveal F(r) and provide a proof that the revealed evaluation is indeed consistent with the committed polynomial.
A Multilinear Polynomial Commitment enables the prover to commit to a multilinear polynomial 'F' in 'k' variables. Similar to the Polynomial Commitment, if the verifier later requests to learn an evaluation of the committed polynomial at a specific point 'r', the prover can reveal the requested evaluation and prove that it is indeed consistent with the polynomial that was committed.
A Vector Commitment allows the prover to commit to a vector of length 'd'. If the verifier later asks the prover to reveal a specific entry of the vector, such as the 'i'th entry of the committed vector 'u_i', the prover can reveal the 'i'th entry of that committed vector and provide a short proof that the revealed value 'u_i' is indeed consistent with the committed Vector. In essence, a Vector Commitment is a functional commitment scheme used by Polynomial Commitments and Multilinear Polynomial Commitments.
Merkle Trees: Transforming to Polynomial Commitments
Now, let's take a step back and try to relate Merkle Trees to Polynomial Commitments. One could see how a Merkle tree is used as a commitment scheme for vectors, but how can we use a similar structure for Polynomial Commitments?
Remember that a Polynomial Commitment allows a prover to commit to a polynomial 'F', and later reveal the evaluation of that polynomial at a specific point 'r', and prove that the revealed evaluation is indeed consistent with the committed polynomial. A natural way to extend this idea to Merkle trees would be to have the polynomial coefficients as the vector elements that are being committed in the Merkle tree.
If we assume the polynomial F to be of the form 'F(x) = a_0 + a_1 * x + a_2 * x^2 + ... + a_d * x^d', the coefficients a_0, a_1, ..., a_d can be viewed as a vector that the prover commits to using a Merkle tree. Later, when the verifier requests to know the evaluation of the polynomial at a specific point 'r', the prover can evaluate the polynomial at 'r' and provide the result, along with a proof (authentication information) that the revealed evaluation is indeed consistent with the committed polynomial.
This transformation from a Merkle tree to a Polynomial Commitment might seem straightforward, but it has its own challenges and intricacies. In the case of Polynomial Commitments, one of the main challenges lies in the construction of the authentication information. It involves extending the concept of siblings in the Merkle tree to a polynomial setting, a concept that is not directly applicable. This topic can be explored further in the study of polynomial and multilinear polynomial commitment schemes.
Merkle Trees: Transforming to Multilinear Polynomial Commitments
Extending the transformation to Multilinear Polynomial Commitments is even more complex. A Multilinear Polynomial Commitment enables the prover to commit to a multilinear polynomial 'F' in 'k' variables. Similar to the Polynomial Commitment, if the verifier later requests to learn an evaluation of the committed polynomial at a specific point 'r', the prover can reveal the requested evaluation and prove that it is indeed consistent with the polynomial that was committed.
The transition from Merkle trees to Multilinear Polynomial Commitments involves additional complexities. As opposed to univariate polynomials, a multilinear polynomial contains multiple variables, which further complicates the authentication information construction and proof verification. The conceptual leap from siblings in a Merkle tree to a multivariate setting presents a nontrivial challenge. Nevertheless, such a transformation is feasible and can be studied in detail under the domain of Multilinear Polynomial Commitments.
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Merkle Trees: Applications
Merkle trees and their extensions, Polynomial Commitments, and Multilinear Polynomial Commitments have wide-ranging applications in cryptography and data integrity. They play a crucial role in various cryptographic protocols including, but not limited to, blockchain technology, proof of storage, privacy-preserving computation, and secure multi-party computation.
The value of these commitment schemes lies in their ability to allow a prover to commit to a set of data (vector, polynomial, etc.), and then subsequently provide a succinct proof for specific properties of the committed data, such as an element of a vector, or an evaluation of a polynomial, while ensuring both data integrity and the prover's commitment.
A First Polynomial Commitment
The concept of Polynomial Commitment, on its surface, is quite simple. In the context of cryptography, a prover wants to convince a verifier that they have knowledge of a certain polynomial, without revealing the polynomial itself.
A first, naive attempt at creating a polynomial commitment scheme might involve a degree d polynomial over a finite field. Let's consider an example with a finite field of seven elements, although in practice, this field would be much larger. We'll denote the polynomial that the prover wants to commit to as F, and it's degree as d.
To commit to this polynomial, we could consider the vector containing all evaluations of F. That is, for each of the seven elements in the finite field (0 through 6 in our example), the vector would contain an entry corresponding to the evaluation of F at that field element. In this way, the prover can create a Merkle commitment to this vector, effectively committing to the polynomial F.
However, this simple scheme is not without its flaws. The two primary issues are related to the prover's computational load and the verifier's inability to guarantee the committed polynomial's degree.
Reveal f(4)
Assuming that the prover has committed to the polynomial F using the vector of all evaluations, let's consider a case where the verifier wants to know the value of F at field element 4. In theory, the prover could reveal that evaluation by providing a Merkle authentication proof for the fifth leaf in the tree (corresponding to F of 4).
However, this brings us to the first issue with our simple polynomial commitment scheme. Specifically, the prover has to compute and store a leaf in the Merkle tree for each element of the field over which the polynomial is defined. For large fields, which are often used in cryptographic applications (e.g., fields of size 2^64 or greater), this can become computationally expensive for the prover, even though the verifier's computational load remains manageable.
What we would ideally like is a prover whose runtime is proportional to the degree of the committed polynomial, not the size of the field. In this way, the prover's computation would be significantly more efficient.
The second issue is even more fundamental. Our current scheme provides no guarantees to the verifier that the vector committed to by the prover is indeed the evaluation vector of a degree d polynomial. The Merkle commitment binds the prover to some vector but offers no insight into the vector's structure.
For our scheme to truly be a polynomial commitment scheme, the prover would need to convince the verifier that all leaves of the tree are evaluations of a degree d polynomial, not just an arbitrary set of field elements.
Addressing these issues will be a crucial aspect of designing a more robust and efficient polynomial commitment scheme. As we will see in later discussions, there are indeed methods that can ensure that the committed function is of the correct degree, and the runtime for the prover is proportional to the degree of the polynomial, not the size of the domain.
Conclusion:
Succinct Non-interactive Arguments of Knowledge, or SNARKs, stand at the forefront of modern cryptography, providing an avenue for secure and efficient proof systems. This article has walked you through the intricacies of SNARKs, from their inception to their real-world applications. We've explored how SNARKs evolve from interactive proofs, dove into the depths of polynomial and multilinear polynomial commitments, and highlighted the critical role of Merkle trees in developing these commitments.
However, as with any complex concept, understanding SNARKs and their related constructs is not without its challenges. We've pointed out the computational issues that arise from naive attempts at creating polynomial commitment schemes and discussed the problems of degree verification and prover efficiency. These hurdles, while daunting, are not insurmountable and present exciting opportunities for future research and innovation in the field of cryptography.
The world of cryptography is constantly evolving, and SNARKs are just one of the many tools being refined and developed to secure our digital landscapes. As our reliance on these systems grows, so does the importance of understanding the mechanisms behind them. We hope this article has enlightened you on the world of SNARKs and sparked your curiosity to explore further. The journey into cryptography is as fascinating as it is challenging, offering a vista into the critical aspects of securing our digital world.
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