Multi Protocol Commitments address the following important requirements:
- How the tagged
mpc::Commitmenthash, committed in Bitcoin Blockchain according toOpretorTapretschemes, is constructed. - How state changes associated with more than one contract can be stored in a single commitment.
The preceding points are addressed through an ordered merkelization of the multiple contracts (actually their transition bundles IDs) in an MPC Tree whose properties will be addressed in depth in this section. Eventually, the root of the tree (mpc::Root) is hashed once more to get the mpc:Commitment which is finally committed in an output of the witness transaction using the appropriate Deterministic Bitcoin Commitment construction.
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Each RGB contract has a unique position in the MPC Tree determined by a modular division applied to its ContractId according to the width of the tree. In this example, the MPC tree has a width of 8.
The commitment of the MPC tree - which goes either into Opret or into Tapret commitments - is the mpc::Commitment constructed in BIP-341 fashion as follows:
mpc::Commitment = SHA-256(SHA-256(mpc_tag) || SHA-256(mpc_tag) || mpc::Root )
Where:
mpc::Rootis the root of the MPC tree whose construction is explained in the following paragraphs.mpc_tag = urn:ubideco:mpc:commitment#2024-01-31follows RGB tagging conventions.
In order to construct the MPC tree we must deterministically find a unique leaf position for each contract, thus:
By setting C the number of contracts and i = {0,1,..,C-1} and by having a ContractId(i) = c_i to be included in the MPC, we can construct a tree with w leaves with w > C (corresponding to a depth d such that 2^d = w), so that each contract identifier c_i representing a different contract is placed in a unique position pos(c_i) determined as a modulus operation detailed below.
In essence, the construction of a suitable tree of width w that hosts each contract c_i in a unique position represents a kind of mining process. The greater the number of contract C, the greater should be the number of leaves w. Assuming a random distribution of pos(c_i), as per Birthday Paradox, we have ~50% probability of a collision occurring in a tree with
In order to avoid too large MPC trees and assuming that the occurrence of collisions is a random process, an additional optimization has been introduced.
The actual formula for determining the leaf position of the contract is:
pos(c_i) = c_i + cofactor mod w pred
Where cofactor is a random number of 16 bytes that can be chosen as a "nonce" to obtain distinct values of pos(c_i) with w fixed. The tree construction process starts from the smallest tree such that w > C, then tries a certain number of cofactor attempts. If none of them can produce C distinct positions, w is increased and a new series of cofactor trials is attempted.
Once C distinct positions pos(c_i) with i = 0,...,C-1 are found, the corresponding leaves are populated through a tagged hash constructed in the following way:
tH_MPC_LEAF(c_i) = SHA-256(SHA-256(merkle_tag) || SHA-256(merkle_tag) || b || d || w || 0x10 || c_i || BundleId(c_i))
Where:
merkle_tag = urn:ubideco:merkle:node#2024-01-31is chosen according to RGB conventions on Merkle Tree tagging commitments.b = 1refers to the branching of the leaf which refers to a single leaf node.dis the depth of the MPC tree at the base layer.wis the width of the MPC tree.0x10is the integer identifier of contract leaves.c_iis the 32-byte contract_id which is derived from the hash of the Genesis of the contract itself.BundleId(c_i)is the 32-byte hash that is calculated from the data of the Transition Bundle which groups all the State Transitions of the contractc_i.
For the remaining w - C uninhabited leaves, a dummy value must be committed. To do that, each leaf in position j != pos(c_i) is populated in the following way:
tH_MPC_LEAF(j) = SHA-256(SHA-256(merkle_tag) || SHA-256(merkle_tag) || b || d || w || 0x11 || entropy || j )
Where:
merkle_tag = urn:ubideco:merkle:node#2024-01-31is chosen according to RGB conventions on Merkle Tree tagging commitments.b = 1refers to the branching of the leaf which refers to a single leaf node.dis the depth of the MPC tree at the base layer.wis the width of the MPC tree.0x11is the integer identifier of entropy leaves.entropyis a 64-byte random value chosen by the user constructing the tree.
After generating the base of the MPC tree having w leaves, merkelization is performed following the rule of commit_verify crate detailed here.
The following diagram shows the construction of an example MPC tree where:
C = 3number of contracts to place.- As an example:
pos(c_0) = 7, pos(c_1) = 4, pos(c_2) = 2. BUNDLE_i = BundleId(c_i).tH_MPC_BRANCH(tH1 || tH2) = SHA-256(SHA-256(merkle_tag) || SHA-256(merkle_tag) || b || d || w || tH1 || tH2).merkle_tag = urn:ubideco:merkle:node#2024-01-31is chosen according to RGB conventions on Merkle Tree tagging commitments.bis the branching of the tree merkelization scheme. In this case it isb = 2meaning that the merkelization happens with 2 input nodes:tH1andtH2, both having a 32-byte length.dis the tree depth which is updated at each level of the tree encoded a an 8-bit Little Endian unsigned integer. The depth at the base of the MPC tree in the example isd = 3wis a 256-bit Little Endian unsigned integer representing the width of the tree which remains fixed in each merkelization. In the example we have:w=8.
{% code fullWidth="true" %}
+--------------------------+
mpc:Root | th_MPC(tHABCD || tHEFGH) |
+-----------^---------^----+
| |
+------------------------------------------------+ +--------------------------------------------+
| |
+-------------+---------------+ +-------------+---------------+
| tH_MPC_BRANCH(tHAB || tHCD) | | tH_MPC_BRANCH(tHEF || tHGH) |
+----------------^--------^---+ +-----------------+--------+--+
| | | |
+---------------------------+ +--------------+ +------------------<---------+ +------------+
| | | |
+-------------+-------------+ +-------------+-------------+ +-------------+-------------+ +-------------+-------------+
| tH_MPC_BRANCH(tHA || tHB) | | tH_MPC_BRANCH(tHC || tHD) | | tH_MPC_BRANCH(tHE || tHF) | | tH_MPC_BRANCH(tHG || tHH) |
+----------------^------^---+ +----------------^------^---+ +----------------^------^---+ +----------------^------^---+
| | | | | | | |
+--------------+ +-----+ +--------------+ +- ---+ +--------------+ + ----+ +--------------+ +-----+
| | | | | | | |
+-------+--------+ +---------+------+ +-------+--------+ +---------+------+ +-------+--------+ +---------+------+ +------+---------+ +--------+-------+
| tH_MPC_LEAF(A) | | tH_MPC_LEAF(B) | | tH_MPC_LEAF(C) | | tH_MPC_LEAF(D) | | tH_MPC_LEAF(E) | | tH_MPC_LEAF(F) | | tH_MPC_LEAF(G) | | tH_MPC_LEAF(H) |
+-------------^--+ +-------------^--+ +-------------^--+ +-------------^--+ +-------------^--+ +-------------^--+ +-------------^--+ +-------------^--+
| | | | | | | |
+--------------+-------+ +--------------+-------+ +--------------+----------+ +-----------+---------+ +------------+------------+ +---------+------------+ +---------+------------+ +---------+---------------+
| 0x11 || entropy || 0 | | 0x11 || entropy || 1 | | 0x10 || c_2 || BUNDLE_2 | | 0x11 | entropy || 3 | | 0x10 || c_1 || BUNDLE_1 | | 0x11 || entropy || 5 | | 0x11 || entropy || 6 | | 0x10 || c_0 || BUNDLE_0 |
+----------------------+ +----------------------+ +-------------------------+ +---------------------+ +-------------------------+ +----------------------+ +----------------------+ +-------------------------+
{% endcode %}
From a verifier's perspective, in order to prove the presence of client-side validated data related to some contract c_i collected in BUNDLE_i, only a Merkle Proof pointing at it inside the tree is needed. Because of this, different verifiers of different contracts don't need to have the full view of the Merkle Tree as the builder does, and this, together with the dummy entropy leaves, provides a high degree of privacy. Using the example tree in the diagram above, a verifier of, say, the contract c_2 will receive the following Merkle Proof from the tree builder:
{% code fullWidth="true" %}
+-------------------------------+
| tH_MPC_ROOT(tHABCD || tHEFGH) |
+----------------^---------^----+
| |
+-----------------------------------------------------+ +---------------------------------------+
| |
+-------------+---------------+ +-------------+---------------+
| tH_MPC_BRANCH(tHAB || tHCD) | | tH_MPC_BRANCH(tHEF || tHGH) |
+----------------^--------^---+ +-----------------------------+
| |
+---------------------------+ +--------------+
| |
+-------------+-------------+ +-------------+-------------+
| tH_MPC_BRANCH(tHA || tHB) | | tH_MPC_BRANCH(tHC || tHD) |
+---------------------------+ +----------------^------^---+
| |
+--------------+ +- ---+
| |
+-------+--------+ +---------+------+
| tH_MPC_LEAF(C) | | tH_MPC_LEAF(D) |
+-------------^--+ +-------------^--+
| |
+-------------------------+ +------+----+---------+
| 0x10 || c_3 || BUNDLE_2 | | 0x11 | entropy || 3 |
+-------------------------+ +------+--------------+
{% endcode %}
So the Merkle Proof provided to verify the existence and uniqueness of contract commitment in the tree is: 0x11 | entropy || 3, tH_MPC_BRANCH(tHA || tHB) and tH_MPC_BRANCH(tHEF || tHGH).