feat(merkle_tree): initial batch membership proof implementation

[crStiv]

This PR provides an initial implementation of batch membership proof functionality that was marked as TODO.
Current implementation includes:

• Basic BatchMembershipProof type
• batch_lookup and batch_verify methods
• Basic iterative approach for batch operations

Next steps (will be discussed in a separate issue):

• Optimization of batch verification
• Adding tests
• Adding documentation
• Possible algorithmic improvements

Related TODO: merkle_tree/src/macros.rs:43

[mrain]

Thank you for your contribution!
However, we are seeking a more compact version of the batch proof. The basic implementation of Vec<MerkleTreeProof<T>> can be easily achieved externally, making its impact rather limited.

[alxiong]

Agree, we'd be happy to see a non-trival batched proof that's shorter than concatenation of n original proofs. Through internal nodes reuse or whatnot.

[crStiv]

@mrain
@alxiong

Hey! I've improved the batch proof to make it more compact by sharing common nodes between proofs instead of storing duplicates. Added proper error checks and tests to make sure everything works correctly. This should address the concerns about proof size from the previous comments.

• Implemented CompactBatchProof that reduces memory usage by:

  • Storing shared nodes only once
  • Keeping only differing nodes for individual proofs
  • Reusing common paths during verification

• Added error handling for invalid inputs

• Included comprehensive test coverage

• Added full documentation