Narcoleptic Programmers' Medicine


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    alt text Merkle Mountain Range

    A type of merkle tree that can be visualized as many (perfect) merkle trees which are then combined into 1, by creating a single root from all of their peaks. The rules for making the tree(s) however are rigidly deterministic such that the entire structure depends only on the number of items put into it. When appending leaves, nodes are not updated, only appended. This makes for a minimal amount of total hash computations (~2n), while maintaining the property that no nodes are updated when appending new leaves (nodes are merely added). Because of this, the merkle inclusion proofs are also only appended to meaning that later proofs are sufficient (supersets of) earlier proofs.

    A Golang version of this can be found here.

    These unique properties make it optimal for proving the work of an entire PoW blockchain. A protocol that does this is FlyClient.

    alt text


    MMR data structure as described by Peter Todd: (Reyzin and Yakoubov may have proposed at a similar data structure in 2015)

    I have improved the structure slightly from the above references (by simplifying it). The difference is in the "bagging the peaks" process (used to calculate the merkle-root). Todd's process takes the peaks and creates another merkle tree from them in order to attain a single root. My process is to simply digest the peaks as a single concatenated array to attain a single root.


    1. Using Todd's process it was (by his own admission) very difficult to logically determine leaf position from an inclusion proof.

    2. The getRoot method concatenates the peaks and hashes them on the fly (the root is never persisted). This process is very cheap because there are only ~log(n)/2 peaks.


    npm install merkle-mountain-range
    const { MMR, keccak256FlyHash } = require('merkle-mountain-range')
    let mmr = new MMR(keccak256FlyHash)
    let genesisHash       = Buffer.from('d4e56740f876aef8c010b86a40d5f56745a118d0906a34e69aec8c0db1cb8fa3','hex')
    let genesisDifficulty = Buffer.from('0000000000000000000000000000000000000000000000000000000400000000','hex')
    let zeroithLeaf = Buffer.concat([genesisHash, genesisDifficulty])
    mmr.append(zeroithLeaf, 0).then(()=>{ console.log(mmr) })


    With 1000 64-byte leaves (2015 macbookpro)

    • MemoryBasedDb
      • Time per append() = 0.000119s
      • Time per get() = 0.000289s
      • Time per mmr._getNodeValue(MMR.getNodePosition(leafIndex)) about 0.00002
    • FileBasedDb
      • Time per append() = 0.000301s
      • Time per get() = 0.000719s
      • Time per mmr._getNodeValue(MMR.getNodePosition(leafIndex)) about 0.00004
    • LevelDbBasedDb
      • Time per append() = 0.000384s
      • Time per get() = 0.000832s

    The cost of mmr.get(leafIndex) can be reduced by instead using mmr._getNodeValue(MMR.getNodePosition(leafIndex)). Because get() verifies as it traverses down the tree. Makes it easy to not mess up verification. You can technically get a leaf much faster with a single read (that does not verify) by calculating the position and reading it directly (O(1) instead of O(logn)).


    How to contribute, build and release are outlined in, and respectively. Try to follow the specification.

    Using semantic versioning (more info here)

    Testing uses mocha. It should work to simply pull down the repo, do an npm install, and use npm run test to run the tests.

    Changelog V0 -> V1

    • .mmr file has slightly different (incompatible) format to store wordsize (which is no longer locked to 64 bytes)
    • FileBasedDb must now be created using .open or .create. .open takes an additional wordSize argument (default = 64 bytes)
    • Support for serialized proof tree (or sparse tree)
    • Digests has been deprecated (now _Digests) because it doesn't belong in this repo. You should include your own hash function package. If it does not have the required function signature - i.e: <buffer> in <buffer> out, then you will have to wrap in in one that does before using it. The digest functions used in Flyclient will be available through a seperate "flyclient" npm package.

    • Added support for LevelDb based databases with optional "key prefix" for data name spacing

    Merkle Proofs & Serialization Format

    The most elegant way of dealing with merkle-proofs is to basically think of them as a sparse tree in which only some of its nodes are present. As long as while performing a get the software walks from peak to leaf checking each hash properly matches its children along the way, this proves inclusion before returning the leaf. If any node required in this path traversal is not present, the sparse tree is insufficient to prove this particular leaf (software currently throws Missing node error); If any hash does not match its parents, the proof is fraudulent (throws Hash mismatch error).

    The merkle Proof data is essentially the same data of a regular MMR (leafLength and nodes) except with only some of the nodes (rather than of all of them). For example the data of a sprase mmr might look like this:

    leafLength: 34
        30 : <12 34 56 78 90>,
        32 : <12 34 12 34 34>,
        33 : <21 43 65 87 09>

    (whereas a similar full mmr of length 34 would contain all the nodes from 0-33)

    To serialize this data for remote transmission we first arrange it into very specifically arranged arrays (of byte arrays) for RLP encoding. On the other end, the software knows that the zeroith item is leafLength and the first item is an array of nodes (in which each node is of length 2 to represent the node key pairs):

      [ <1e>, <12 34 56 78 90> ],
      [ <20>, <12 34 12 34 34> ],
      [ <21>, <21 43 65 87 09> ]

    Finally we can rlp.encode() this data:

    <d9 22 c7 1e 85 12 34 56 78 90 c7 20 85 12 34 12 34 34 c7 21 85 21 43 65 87 09>

    rlp.decode() gets you exactly back to the above arrays, and the software can reinterpret the items as described above to rebuild the sparse tree. Note that these serialized bytes can be passed from Golang package to JS package and vise-versa!

    NOTES (mostly to self):

    there probably should be a whole framework for requesting more nodes (to complete insufficient proofs). A few ideas on this.

    • somehow the verifier must get the diffof node positions that is has, vs ones that it needs.
    • it would be maximally flexible to be able to request individual nodes - so we need a getNodes(nodeIndexes) method that the prover can use to return them.
    • however it would often be more concise to request particular leafIndexes and the prover would calculate which nodes to return to satisfy that. Of course this would likely return some nodes the verifier already has (like always the peaks).
    • It would be useful for these requests to include the leafLength AND the root which would LOCK the prover to an EXACT response any variation from which would be known to be fraudulent (without this, the prover could return nodes from some different tree that it genuinely thinks you are talking about (for instance a different micro-fork/re-org in our blockchain context)).


    npm i @subql/x-merkle-mountain-range

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