Merkle Trees: A Fundamental Structure in Cryptography

Merkle trees, introduced by Ralph Merkle in 1979, are a pivotal data structure in cryptographic systems. These binary hash trees enable efficient and secure verification of data integrity within distributed systems. Their design capitalizes on hash functions to reduce computational overhead, making them indispensable in blockchain and peer-to-peer networks.

What are Merkle Trees?

Structure and Construction

A Merkle tree is a rooted binary tree where:

  • Leaf nodes store the hash of individual data blocks:$h_0 = H(d_0), h_1 = H(d_1), \dots, h_n = H(d_n)$, where $H$ is a cryptographic hash function, and$d_i$represents a data block.
  • Internal nodes are computed iteratively by concatenating and hashing child nodes: $h_{\text{parent}} = H(h_{\text{left child}} || h_{\text{right child}})$.
  • The root node, or Merkle root, encapsulates the integrity of the entire dataset.

For$n$leaf nodes, the tree contains$2n - 1$nodes in total. Efficient construction and traversal are enabled by this logarithmic height.

Proof of Inclusion

Merkle trees facilitate compact proofs of inclusion for a data block$d_i$without requiring access to the full dataset:

  1. A Merkle proof includes the hashes of sibling nodes along the path from$d_i$to the root.
  2. The verifier iteratively reconstructs the root from$d_i$and the provided proof, comparing it with the stored root.

The computational cost of verification is $O(\log n)$ , where $n$ is the number of data blocks.

Security Properties

Merkle trees derive their robustness from cryptographic hash functions, see Tabelle di hash

Collision and Preimage resistance

  • Collision resistance ensures that altering any$d_i$requires recalculating the entire Merkle root, making tampering detectable.
  • Preimage resistance prevents reversing the hash, ensuring data confidentiality.

Probability of corruption

If the dataset contains $n$ blocks, the probability of undetected corruption is negligible given a strong hash function $H$. For a hash function with $b$ -bit output, the collision probability is approximately $\frac{1}{2^b}$, emphasizing the importance of selecting secure hash algorithms like SHA-256 or SHA-3.

Applications

Merkle trees underpin many systems, including:

  1. Blockchain Networks:
    • Bitcoin and Ethereum employ Merkle trees to validate transactions efficiently within blocks.
    • Light clients rely on Merkle proofs to confirm transactions without downloading the entire blockchain.
  2. File Systems:
    • Distributed file systems like IPFS use Merkle trees for data versioning and deduplication.
  3. Certificate Transparency:
    • Logs of public key certificates are stored and audited using Merkle trees.
    • They have also been used for Cloud Storage#Key-value stores in the dynamo replication algorithm to check for inconsistencies in the replicas.