Matematicka Analiza Merkle 19.pdf Jun 2026

The tree structure ensures that any change to a single data block changes ( R ) with overwhelming probability (collision resistance of ( H )).

Most introductions to Merkle trees stop at the pretty picture: a binary tree where leaves are data blocks, and the root is a single fingerprint of everything below. But a mathematical analysis asks the brutal questions:

$$\textMinimize D(b) = \lceil \log_b N \rceil \cdot \left( C_\texthash \cdot b + C_\textnet \right)$$ Matematicka Analiza Merkle 19.pdf

It appears this may be:

Let us define a binary Merkle tree over a set of data blocks ( D = d_1, d_2, \dots, d_n ) where ( n = 2^k ) for simplicity (padding with dummy data if needed). The tree structure ensures that any change to

If ( H ) is collision-resistant, an adversary cannot forge a proof for ( d'_i \notin D ) to match ( R ). Proof: A false proof would imply a collision at the first differing internal node.

The single hash value at the top of the tree after recursively computing all internal nodes: ( R = H(\dots H(H(L_1|L_2)) \dots) ). If ( H ) is collision-resistant, an adversary

That is a 75% reduction in proof size. But here is the mathematical twist: the cost of hashing a node with 19 children is not linear. The analysis likely dissects the . You save on network rounds (good for latency), but you pay in local computation (hashing a wide node takes more CPU).

Where $b$ is the branching factor, $C_\texthash$ is the cost of hashing one child, and $C_\textnet$ is the cost of transmitting one hash.

The analysis might reveal a : For branching factors below 19, the tree is robust; above 19, certain algebraic attacks (using the pigeonhole principle on intermediate nodes) become statistically viable.