zk-SNARKs: Succinctness, Fast Verification, and the Trusted Setup Dilemma

Introduction

Zero-Knowledge Succinct Non-Interactive Argument of Knowledge (zk-SNARKs) have gained significant attention in recent years due to their remarkable properties: succinctness, fast verification, and zero-knowledge proofs. zk-SNARKs have the potential to revolutionize various application domains, including cryptocurrencies, identity verification, and secure voting systems. However, the Trusted Setup phase, which is a critical component of zk-SNARKs, poses a significant security risk. In this blog post, we will delve into the intricacies of zk-SNARKs, their advantages, and the Trusted Setup dilemma.

Succinctness and Fast Verification

zk-SNARKs are designed to provide a proof that a statement is true without revealing any information about the statement itself. This is achieved by using a commitment scheme, which binds the statement to a public key. The proof is then generated by computing a hash function over the statement and the public key. The resulting hash value is used to demonstrate that the statement is true without revealing any information about the statement.

The succinctness of zk-SNARKs is due to the use of a commitment scheme, which reduces the size of the proof. The proof is typically around 1 kilobyte (kB) in size, making it extremely compact. This compactness is essential for many applications, such as cryptocurrencies, where transaction data needs to be transmitted efficiently.

Fast verification is another key property of zk-SNARKs. The verification process involves checking the proof against the public key and the statement. This process is typically performed in milliseconds, making it fast and efficient.

zk-SNARKs Algorithms

The zk-SNARKs algorithm consists of the following components:

• Commitment scheme: A commitment scheme is used to bind the statement to a public key. The commitment scheme is typically based on a hash function.
• Hash function: A hash function is used to compute the proof. The hash function is designed to be collision-resistant and pre-image resistant.
• Public key: The public key is used to verify the proof. The public key is typically generated using a key generation algorithm.

Here is an example of a zk-SNARKs algorithm in Python:

import hashlib

def commit(statement, public_key):
    # Compute the commitment
    commitment = hashlib.sha256((statement + public_key).encode()).hexdigest()
    return commitment

def prove(statement, public_key, commitment):
    # Compute the proof
    proof = hashlib.sha256((statement + commitment).encode()).hexdigest()
    return proof

def verify(proof, public_key, statement):
    # Verify the proof
    commitment = commit(statement, public_key)
    if proof == hashlib.sha256((statement + commitment).encode()).hexdigest():
        return True
    return False

Trusted Setup Dilemma

The Trusted Setup phase is a critical component of zk-SNARKs. During this phase, a trusted party generates a Common Reference String (CRS) that is used to verify the proofs. The CRS is designed to be publicly accessible, allowing anyone to verify the proofs.

However, the Trusted Setup phase poses a significant security risk. If the trusted party is malicious and retains the setup parameters, the security of the entire system can be compromised. This is because the trusted party can use the setup parameters to generate fake proofs that can be used to deceive the verification process.

Quantum Attacks

Another significant issue with zk-SNARKs is their vulnerability to quantum attacks. zk-SNARKs based on Elliptic Curve Cryptography (ECC) are particularly vulnerable to quantum attacks, as ECC is based on the difficulty of the elliptic curve discrete logarithm problem (ECDLP). The ECDLP is considered to be vulnerable to quantum attacks, as quantum computers can efficiently solve the problem.

Best Practices

To mitigate the Trusted Setup dilemma, it is essential to use a secure and auditable setup process. This can be achieved by using a decentralized setup process, where multiple parties are involved in generating the CRS. Additionally, the setup process should be designed to prevent the trusted party from retaining the setup parameters.

To mitigate quantum attacks, it is essential to use a post-quantum secure protocol. One option is to use a lattice-based protocol, which is resistant to quantum attacks.

Conclusion

zk-SNARKs are a powerful cryptographic primitive that offers succinctness, fast verification, and zero-knowledge proofs. However, the Trusted Setup dilemma and quantum attacks pose significant security risks. To mitigate these risks, it is essential to use a secure and auditable setup process and a post-quantum secure protocol. By following best practices and using advanced cryptographic techniques, we can ensure the security and integrity of zk-SNARKs-based applications.