Comparison of Different Proof Schemes: Understanding the Advantages and Disadvantages of ZK Proof Systems

Author: Hill.bit

Compiled by: Deep Tide TechFlow

The concept of zero-knowledge proofs is well-known, but many people may be confused about the technical details.

Zero-knowledge and proof are actually two terms, and the proof scheme is a fundamental component of the security assumptions of zero-knowledge protocols. In this article, Hill.bit will help more people understand ZK proof systems by explaining various proof schemes and their advantages and disadvantages.

In a zero-knowledge proof system, there are three entities involved: the setup, the prover, and the verifier. Different proof schemes will affect their behavior in various ways, thus impacting efficiency, security, and the overall performance of the system.

The setup phase generates the necessary parameters and public keys required for the ZK system. The proof scheme affects the complexity, computation, communication of the setup phase, and whether it is trusted or trustless. The prover generates a proof that demonstrates they possess information about a secret input without revealing that information. The proof scheme affects the prover's computation time, memory requirements, and proof size, thereby influencing communication and storage needs. The verifier checks the validity of the proof. The proof scheme will affect the verification time, memory requirements, and the number and complexity of proof requests. Here are three different types of proof schemes.

Linear PCPs + Only Linear Encoding:

• Utilizes linear probabilistically checkable proofs (PCPs) and linear operations;

• Provides strong zero-knowledge properties;

• Generates the shortest proofs;

• Requires a trusted setup;

• Previous improvements have mainly focused on reducing prover time.

Linear PCPs are a proof system where the verifier checks the validity of a statement by querying a small number of proofs. The term "linear" refers to the fact that the verifier's queries are linear functions of the proof.

Only linear encoding is a cryptographic technique used to hide information, allowing only linear operations on the hidden data. This ensures data privacy while enabling certain computations to be performed.

Polynomial IOPs + Polynomial Commitment Schemes:

• Utilizes algebraic structures;

• Generally more efficient than systems based on linear PCPs;

• Supports universal/trustless setups;

• Allows for customized circuits;

• Previous improvements have mainly focused on enhancing verifier efficiency.

Polynomial Interactive Oracle Proofs (IOPs) are a proof system where the prover and verifier exchange messages over multiple rounds. The prover generates an oracle (a commitment to a polynomial) and provides it to the verifier.

The verifier queries the oracle at specific points, while the prover responds with evaluations of the corresponding polynomial. The polynomial scheme commits to the polynomial without revealing information about the polynomial itself.

Efficiency improvements compared to Linear PCPs + Only Linear Encoding come from:

• Better utilization of algebraic structures;

• More efficient proof generation/verification;

• Compressed polynomial representations;

• Batch verification techniques.

However, Polynomial IOPs + Polynomial Commitment Schemes have the following drawbacks:

• More complex design and implementation;

• Specific-purpose cryptographic assumptions;

• Different performance trade-offs, such as parallelizability.

Recursive Schemes:

• Allow recursive proof composition;

• Implement nested proofs for improved efficiency and scalability;

• Fast and easily parallelizable provers;

• Previous improvements have mainly focused on building recursive SNARKs.

Recursive proof composition can reduce the computational and memory requirements for the verifier, which is particularly useful in applications like blockchain. Proof aggregation can reduce the size and verification time of the final proof, but generating such proofs may impose higher computational requirements on the prover. Compared to Polynomial IOPs + Polynomial Commitment Schemes, efficiency improvements in Recursive Schemes come from:

• Recursive proof composition;

• Proof aggregation;

• Improved scalability;

• Faster verification times.

Potential drawbacks of Recursive Schemes include:

• More complex design and implementation;

• Customized cryptographic assumptions;

• Increased computational time and memory overhead for the prover;

• Applicability may vary depending on the use case.

In summary, Linear PCPs + Only Linear Encoding offer strong zero-knowledge properties and the shortest proof lengths, but they require a trusted setup and have limitations in efficiency compared to other categories. Polynomial IOPs + Polynomial Commitment Schemes show significant improvements in efficiency through more efficient proof generation and verification processes, but their design and implementation may be more complex.

Recursive Schemes excel in efficiency and scalability, benefiting from recursive proof composition, which is particularly useful in blockchain applications. However, the computational time and memory overhead for the prover may increase, and their applicability may vary depending on the use case.