https://x.com/darex_1010/status/1829810989271040352

General Information about Verkle Trees

Verkle Trees are a cryptographic accumulator data structure that allows for compact proofs of inclusion (or non-inclusion) of elements in a set. They are an extension of Merkle Trees but use vector commitments (specifically polynomial commitments) instead of simple hash functions to allow for smaller proof sizes. This efficiency in proof size and the ability to maintain privacy while proving statements makes them a strong candidate for blockchain and other cryptographic applications where space and verification time are critical.

Verkle Trees are particularly relevant in blockchain technology for reducing the size of proofs required to verify the state of the network, which is crucial for scalability and performance in decentralized systems. They offer a balance between proof size and verification efficiency, which is why they are being considered as a future upgrade to existing blockchain protocols like Ethereum.

Summary of the Thread on Verkle Tree and Related Challenges

The discussion centers on the use of Verkle Tree, a data structure that is gaining attention due to its potential efficiency improvements over Merkle Trees in blockchain systems. Verkle Trees leverage a mathematical structure known as a polynomial commitment, which, in this case, relies on the Banderwagon curve—a curve defined over the scalar field of BLS12-381, a pairing-friendly elliptic curve.

Key Points:

1. Verkle Trees vs. Merkle Trees:
   • Unlike Merkle Trees, where the proof size increases with the width of the trie, Verkle Trees offer a proof size that depends primarily on the depth of the trie.
   • This makes Verkle Trees more efficient in terms of proof size as the trie grows.
2. Challenges with Verkle Tries:
   • Multi Scalar Multiplication (MSM): The prover must perform MSM operations to fold multiple Pedersen-style commitments, which are computationally intensive. Optimizing these MSM operations is crucial, as the cost increases with the number of terms involved.
   • Non-Native Arithmetic: A significant challenge arises from the need to perform arithmetic in the Banderwagon scalar field, which is not natively supported by many zk-SNARK systems that are optimized for BLS12-381. This necessitates finding efficient ways to perform these non-native operations within the proof system.
3. Optimization Techniques:
   • Several advanced algorithms like Stratus method, Pippenger's algorithm, and batch-inversion techniques are mentioned for optimizing MSM.
   • For non-native arithmetic, potential solutions include precomputed lookup tables, casting, and limb decomposition. However, these add complexity and potential performance overhead to the proof generation and verification process.
4. Comparisons and Potential Solutions:
   • The thread compares Verkle Trie challenges with issues faced in recursive zk-SNARK systems like Plonky2 and Halo, where similar non-native arithmetic problems arise.
   • Possible solutions include encoding arithmetic operations directly on the elliptic curve, which can be computationally expensive but may provide a workaround for non-native field challenges.
5. Field Size and Performance:
   • The size of the finite field used by the prover is another bottleneck. Larger fields like those used in BLS12-381 may not be as performant as smaller fields used in other zk-SNARK systems, though Banderwagon lacks the advantageous structure of these smaller fields.
   • There’s a consideration of using a hybrid approach, where BLS12-381 is used for verifying Verkle proofs, and smaller fields are used for other zkEVM logic.