Proof of Inclusion

Proof of Inclusion is a concept used to proved that a specific element is part of a set or data structure without revealing the entire set or data structure.

It provides a way to verify the membership of an element efficiently and securely.

One popular application of Proof of Inclusion is in Merkle Trees, which are binary hash trees used in blockchain technology and other distributed systems. In a Merkle Tree, each leaf node represents a data element, and the internal nodes are the hashes of their child nodes. The root of the tree represents the overall hash value of the entire data set.

To prove the inclusion of a specific element in a Merkle Tree, a Proof of Inclusion provides a minimal set of cryptographic hashes or proofs that allows anyone to verify the element's presence without revealing the entire tree. The proof typically consists of the hashes along the path from the leaf node containing the element up to the root of the tree.

Untitled

By verifying the hashes in the Proof of Inclusion against the known root hash, it is possible to confirm the inclusion of the element in the Merkle Tree.

If the hashes match, it provides cryptographic evidence that the element is part of the original data set.

Proof of Inclusion enables efficient and secure verification of data integrity and membership without the need to transmit or store the entire data structure. It is widely used in blockchain systems to validate transactions and ensure the integrity of stored data.

Merkle Root Hash

The Block Header includes a merkle root hash that goes at follows:

Bytes	Name	Data Type	Description
32	merkle root hash	char[32]	A SHA256(SHA256()) hash in internal byte order. The merkle root is derived from the hashes of all transactions included in this block, ensuring that none of those transactions can be modified without modifying the header. See the merkle trees section below.

Merkle Tree per Bitcoin Core Documentation

The merkle root is constructed using all the TXIDs of transactions in this block, but first the TXIDs are placed in order as required by the consensus rules:

The coinbase transaction’s TXID is always placed first.
Any input within this block can spend an output which also appears in this block (assuming the spend is otherwise valid). However, the TXID corresponding to the output must be placed at some point before the TXID corresponding to the input. This ensures that any program parsing block chain transactions linearly will encounter each output before it is used as an input.