Implementing Merkle Trees in Rust 💻

In this section, we'll explore the practical implementation of Merkle trees in Rust using visual intuition. We’ll cover:

1. Calculating the Merkle Root Recursively 🌳
2. Generating a Proof 🔍
3. Validating a Proof ✅
4. Working with Compact Multiproofs 📦

1. Calculating the Merkle Root Recursively 🌳

The Merkle root is the top-most node of a Merkle tree, representing the combined hash of all the data blocks. Let's break down how you can calculate it recursively.

Recursive Merkle Root Calculation

Visual Steps:

• Step 1: Start with your data blocks and hash each one to create the leaf nodes.

  Data Blocks: [D1, D2, D3, '']  ← Pad the base layer if the number of data is not N**2
  Leaf Nodes:  [H1, H2, H3, H4]  ← Hash each data block

• Step 2: Combine each pair of hashes to create the next level of the tree.

      Level 1:  [H1, H2, H3, H4]
      Combine:  H1 + H2 = A,  H3 + H4 = B
      Hash the combination: [Hash(A), Hash(B)]  ← List it to the parent level hashes

• Step 3: Recursively repeat this process until you have a single hash, the Merkle root.

  Final Level:
        Root
         |
         A + B = Root Hash

Visual Example

Here's what the recursive process looks like:

  Level 1:   H1    H2    H3    H4    ← Hashes of data blocks
              \   /      \   /
  Level 2:   Hash(A)     Hash(B)           ← Combine hashes to form next level
                 \     /
  Root Level:     Root Hash          ← Combine recursively to get the final Merkle root

2. Generating a Proof 🔍

A Merkle proof allows you to verify that a specific data block is part of a Merkle tree. Let’s visually explore how to generate this proof.

Generating a Proof

Visual Steps:

• Step 1: Identify the path from the target leaf node (your data block) to the root.

  • Example: To prove that H2 is part of the tree:

  Path: H2 → A → Root

• Step 2: Collect the hashes of the sibling nodes along this path. For each node, check the position of its sibling (either left or right).

  • Example:

  Siblings Needed: H1 (sibling of H2), B (sibling of A)

• Step 3: Add the sibling hashes to a proof list, marking their positions (Left or Right).

  • Example:

  Proof = [Left: H1, Right: B]

Visual Example

Here's a diagram to illustrate generating a proof:

        Root Hash
           /  \
         A      B
        / \    / \
      H1  H2  H3  H4

 Proof = [Left: H1, Right: B]

  Proof for H2:
  1. Start at H2.
  2. Collect sibling H1 (Left).
  3. Move to parent A and collect its sibling B (Right).
  4. Reach the root.

  Final Proof Path: [H1, B]

3. Validating a Proof ✅

After generating a proof, you can validate that a data block is indeed part of the Merkle tree. Here’s how this validation works visually.

Validating a Proof

Visual Steps:

Inputs:

• Original Root Hash
• Data Block Hash: H2
• Proof: [Left: H1, Right: B]

        Root Hash
           /  \
         A      B
        / \    / \
      H1  H2  H3  H4

• Step 1: Start with the hash of the data block you’re validating.

  • Example:

  Start with: H2

• Step 2: Sequentially combine the data block hash with the sibling hashes from the proof, following the order used to create the tree.

  • Example:

  Combine H2 with Left: H1 → A
  Combine A with Right: B → Root

• Step 3: Compare the resulting root hash with the original Merkle root.

  • If they match, the proof is valid, confirming that H2 is part of the tree.

Visual Example

 Proof = [Left: H1, Right: B]

        Root Hash
           /  \
         A      B
        / \    / \
      H1  H2  H3  H4

  Validating H2:
  1. Start with H2.
  2. Combine H2 with H1 (Left) to get A.
  3. Add the hash A to the proof list.
  4. Combine A with B (Right) to get the Root.
  5. Compare the calculated Root with the original Root Hash.
  6. If the roots match, H2 is successfully validated as part of the tree.

4. Working with Compact Multiproofs 📦

Compact multiproofs allow you to efficiently prove the inclusion of multiple data blocks in a Merkle tree. Unlike traditional Merkle proofs, which require padding to make the number of leaves a power of two, compact multiproofs handle arbitrary numbers of leaves without padding. Let’s explore how to generate and validate compact multiproofs.

An example compact merkle tree:

        Root Hash
           /  \
         A      H3
        / \   
      H1  H2

Generating a Compact Multiproof

Visual Steps:

• Step 1: Identify the indices of the data blocks you want to prove.

  • Example:

  We want to prove the inclusion of blocks at indices 0, 1, and 6.
  leaf_indices = [0, 1, 6]

• Step 2: Collect only the necessary sibling hashes required to prove all the selected blocks. This ensures that the proof is as small as possible while still being valid.

  • Example:

  Sibling hashes needed: H1 (for H2), H4 (for H3)

Visual Example

Consider the following Merkle tree:

                                      Root            
                                   /     \           
                                O           O     
                              /   \       /   \     
                             O    H_1   H_2    O  
                            / \   / \   / \   / \
                           X  X  O  O  O  O  X  H_0

For proving the blocks at indices 0, 1, and 6:

• Leaf Nodes: [X, X, X] represent the data blocks at indices 0, 1, and 6.
• Sibling Nodes: Collect the necessary siblings like H_1 and H_2.
• Proof Structure: Combine the collected siblings to form the compact multiproof.

Generating a Compact Multiproof

Proof = [H_0, H_1, H_2]

In this example, the compact multiproof allows for proving multiple blocks with fewer hashes compared to generating separate proofs for each block.

---

Validating a Compact Multiproof

Visual Steps:

• Step 1: Reconstruct the hashes for each subtree from the provided blocks and the proof.

  • Example:

  Combine H1 + H2 to get A
  Combine H3 + H4 to get B

• Step 2: Compare the resulting root hash with the stored Merkle root to validate the proof.

Visual Example

Let’s validate a compact multiproof:

        Root Hash
           /  \
         A      B
        / \    / \
      H1  H2  H3  H4

For validating blocks H1, H2, and H3:

• Step 1: Reconstruct the hash A from H1 and H2.
• Step 2: Reconstruct the hash B from H3 and H4.
• Step 3: Combine A and B to form the root hash.
• Step 4: If the root matches the stored root hash, the proof is valid.