[bitcoin-dev] A Method for Computing Merkle Roots of Annotated Binary Trees

["Russell O'Connor" <roconnor@blockstream.io>]

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## Introduction
This document aims to specify and justify a method for computing Merkle
roots for tree structures whose nodes are annotated with other data.  While
this proposal could be used to replace Bitcoin's Merkle root calculation,
it is primarily aimed at new applications such as MAST, (U)TXO commitments
or other Merklized structures.

## Background
Bitcoin uses a Merkle tree construction to build a commitment to a sequence
of transactions for a Bitcoin block.  The main operation for computing a
Merkle tree is one that takes the recursively computed Merkle roots of two
branches and combines them to compute the Merkle root of the tree with
those two branches.  In Bitcoin, a Merkle root is 256 bits and the
construction is

    MerkleRoot := SHA256(SHA256(LeftRoot ⋅ RightRoot))

One problem with this construction is that it is unnecessarily expensive.
While the concatenation of the LeftRoot and the RightRoot fits in 512 bits,
the size of a SHA256 chunk, a second chunk is needed to fit SHA256's
padding.  This means that inner SHA256 call invokes the SHA256 compression
function twice, once for each chunk.  The outer SHA256 call invokes the
SHA256 compression function a third time.  Bitcoin's Merkle root procedure
calls the SHA256 compression function a total of three times per node.

The main purpose of composing two calls to the SHA256 hash function is to
protect against length extension attacks.  A length extension attack is
where an attacker has access to the hash of a message `HASH(msg)` and,
without knowing `msg`, is able to construct the hash of a new message
`HASH(msg ⋅ attackerMsg)`.  This is attack is usually used against poorly
constructed MACs (Message Authentication Codes).  In the case of Bitcoin
there are no secret messages that are hashed, so the outer call to SHA256
is unnecessary.

As mentioned above, the inner call to SHA256 requires two chunks because of
SHA256's padding.  For the MerkleRoot construction added padding value is
always


0x80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000200

because the input is of a fixed length.  SHA256's padding function is
designed

1. to add bits to the message so that the resulting message size is a
multiple of the chunk size (which is 512-bits in the case of SHA256).
2. to satisfy the Merkle--Damgård property which says that if we can find a
hash collision in SHA256, which operates on variable length messages, then
we can find a hash collision in SHA256's compression function, which
operates on fixed length messages.

We could remove the second call to the SHA256 compression function and use
SHA256 without padding.  If we did, we would lose the Merkle--Damgård
property.  While this may be acceptable for some use cases, we are still
left with no room to add annotation data held at a node.  For Bitcoin's
Merkle tree this is not a problem, because its trees are not annotated.
However, for other applications, we would need to add another chunk to hold
the annotation, which would mean adding back the second call to the SHA256
compression function per node of the tree.

## A New Merkle Root Algorithm
Below is an algorithm for computing Merkle roots that directly uses the
SHA256 compression function.  This algorithm operates on binary trees and
supports per node annotations.  Furthermore this proposal satisfies the
Merkle--Damgård property and more.

#### Remark 1
Any finitely branching tree can be represented by a binary tree, so this
construction also applies to arbitrary finitely branching trees.

The SHA256 compression function takes two inputs:

1. A 256-bit value for the previous chunk in a chain, or an initial vector
in the case of the first chunk.
2. A 512-bit chunk of data.

    sha256Compress : Word256 × Word512 -> Word256

In total, the SHA256 compression function compresses 768-bits into
256-bits.  The Merkle roots of two branches occupy 512 bits, and this
leaves another 256-bits of space available for tags.

Let us define a recursive type for binary trees annotated with tags.

    type Tree tag where
      Leaf   : tag                       -> Tree tag
      Unary  : tag × Tree tag            -> Tree tag
      Binary : tag × Tree tag × Tree tag -> Tree tag

We define a recursive algorithm for trees whose tags are 256-bit Words (or
whose tags can be represented by 256-bit words).

    merkleRoot : Tree Word256 -> Word256
    merkleRoot (Leaf (t))                :=
sha256Compress(sha256(ApplicationName),
0b0^256 ⋅ t)
    merkleRoot (Unary (t, child))        := sha256Compress(merkleRoot(child),
0b0^256 ⋅ t)
    merkleRoot (Binary (t, left, right)) := sha256Compress(merkleRoot(left),
merkleRoot(right) ⋅ t)

We need some initial value for the `Leaf` case.  This could be taken as the
initial vector for SHA256.  However, I suggest using the hash of an
application specific name.

    ApplicationName : BitString

We further require that the tags dictate the kind of node it is attached
to.  For example, if a given tag is used for Binary nodes, then it can only
appear in other Binary nodes.  One way this can be accomplished is by
requiring the first two bits of a tag follow a particular scheme (the
scheme below ensures that one of the first two bits is set, which will be
useful later when we define safe tags) such as

- 0b11 when the node is a Leaf node.
- 0b10 when the node is a Unary node.
- 0b01 when the node is a Binary node.

This condition suffices to provide the Merkle--Damgård property:

#### Theorem 2
Suppose `t_1` and `t_2` are two different `Tree Word256` values such that
any tag occurring in the two trees only occurs in one kind of node.  If
`merkleRoot(t_1) = merkleRoot(t_2)` then we can effectively find a
collision in the SHA256 compression function.

Proof.
We proceed by structural induction on `t_1`.  Assume `t_1` and `t_2` are
two different `Tree Word256` values such that `merkleRoot(t_1) =
merkleRoot(t_2)`.  Define a `tag` function to extract the tag name from the
root of a tree.

    tag : Tree Word256 -> Word256
    tag (Leaf (t))         := t
    tag (Unary (t, _))     := t
    tag (Binary (t, _, _)) := t

Case 1: Suppose `tag(t_1) =/= tag(t_2)`.
This means that the inputs to `sha256Compress` that produced
`merkleRoot(t_1)` and `merkleRoot(t_2)` are different and we have found a
collision in the SHA256 compression function.

Case 2: Suppose `tag(t_1) = tag(t_2)`.
Let `tg` be this tag.  By our requirement on tags, this means that `t_1`
and `t_2` are the same kind of node.  Suppose `t_1 = Binary (tg, t_(1l),
t_(1r))` and `t_2 = Binary (tg, t_(2l), t_(2r))`.  Since `t_1` and `t_2`
are different, then either `t_(1l) =/= t_(2l)` or `t_(1r) =/= t_(2r)`.
Without loss of generality, assume `t_(1l) =/= t_(2l)`.

Case 2a: Suppose `merkleRoot(t_(1l)) =/= merkleRoot(t_(2l))`.
This means that the inputs to `sha256Compress` that produced
`merkleRoot(t_1)` and `merkleRoot(t_2)` are different and we have found a
collision in the SHA256 compression function.

Case 2b: Suppose `merkleRoot(t_(1l)) = merkleRoot(t_(2l))`.
Then by induction we can find a collision in the SHA256 compression
function.

The cases where `t_1` and `t_2` are both `Unary` or `Leaf` nodes are
handled in a similar way.  The reader can verify the algorithm implied by
the above proof provides an effective method of finding a collision in the
SHA256 compression function.  QED

#### Remark 3
We have filled half of the chunk with `0b0^256` in for the `Unary` and
`Leaf `cases in the definition of `merkleRoot`.  The proof of Theorem 2
does not depend on this, and if we have extra ancillary data for these
types of nodes it can replace the `0b0^256` in this first half of the
chunk.  This is particularly useful for the `Leaf `case because `Leaf`
nodes often hold extra data.  We also remark that if this data is too large
to be represented by a `Word256`, it can instead be hashed with SHA256 and
the hash included instead.

Not all of the inputs to the SHA256 compression function are created
equal.  Only the second argument, the chunk data, is applied to the SHA256
expander.  `merkleRoot` is designed to ensure that the first argument of
the SHA256 compression function is only fed some output of the SHA256
compression function.  In fact, we can prove that the output of the
`merkleRoot` function is always the midstate of some SHA256 hash.  To see
this, let us explicitly separate the `sha256` function into the padding
step, `sha256Pad`, and the recursive hashing step, `unpaddedSha256`.

    sha256Pad : BitString -> List Word512

    sha256IV : Word256

    sha256Loop : Word256 × List Word512 -> Word256
    sha256Loop (prev, Nil)                := prev
    sha256Loop (prev, Cons (chunk, rest)) := sha256Loop(sha256Compress(prev,
chunk), rest)

    unpaddedSha256 : List Word512 -> Word256
    unpaddedSha256 (chunks) := sha256Loop(sha256IV, chunks)

    sha256 : BitString -> Word256
    sha256 (s) := unpaddedSha256(sha256Pad(s))

Now we can state what we mean when we say that the output of the
`merkleRoot` function is always the midstate of some SHA256 hash.

#### Theorem 4
For all `t : Tree Word256` there exists some `l : List Word512` such that
`merkleRoot(t) = unpaddedSha256(l)`.

Proof.
We construct a function to transform a tree into the required list of
chunks.

    merkleChain : Tree Word256 -> List Word512
    merkleChain (Leaf (t))                := sha256Pad(ApplicationName) +
[0b0^256 ⋅ t]
    merkleChain (Unary (t, child))        := merkleChain(child) + [0b0^256
⋅ t]
    merkleChain (Binary (t, left, right)) := merkleChain(left) +
[merkleRoot(right) ⋅ t]

The reader can verify by induction that

    forall t : Tree Word256. merkleRoot(t) = unpaddedSha256(merkleChain(t)).

QED

### Large Tag Space

If one's application cannot directly represent the space of tags with a
`Word256`, then one can hash the tags and still maintain the
Merkle--Damgård property given by Theorem 2, as long as we still have the
property that each tag uniquely determines the kind of node it applies to.

We first note that `Tree` is a functor.

    treeMap : (a -> b) -> Tree a -> Tree b
    treeMap f (Leaf (t))                := Leaf (f(t))
    treeMap f (Unary (t, child))        := Unary (f(t), child)
    treeMap f (Binary (t, left, right)) := Binary (f(t), left, right)

We can hash the tags of a tree.

    hash256Tags : Tree BitString -> Tree Word256
    hash256Tags (t) := treeMap sha256 t

Now we can compute the `merkleRoot` of the result of `hash256Tags`.

#### Theorem 5
Suppose `t_1` and `t_2` are two different `Tree BitString` values such that
any tag occurring in the two trees only occurs in one kind of node.  If
`merkleRoot(hash256Tags(t_1)) = merkleRoot(hash256Tags(t_2))`, then we can
effectively find a collision in the SHA256 compression function.

Proof.
Assume `t_1` and `t_2` are two different `Tree BitString` values such that
`merkleRoot(hash256Tags(t_1)) = merkleRoot(hash256Tags(t_2))`.

Case 1: Suppose `hash256Tags(t_1) = hash256Tags(t_2)`.
This means there must be a collision in the `sha256` function.  By the
Merkle--Damgård property of SHA256, this means we can effectively find a
collision in the SHA256 compression function.

Case 2: Suppose `hash256Tags(t_1) =/= hash256Tags(t_2)`.
If the SHA256 hashes of each tag in the two trees uniquely determines the
kinds of their nodes, then we can apply Theorem 2 to conclude that we can
effectively find a collision in the SHA256 compression function.  If the
SHA256 hashes of each tag in the two trees does not uniquely determine the
kinds of their nodes then there are two different tags among the two trees,
`tg_1` and `tg_2`, such that `sha256(tg_1) = sha256(tg_2)`.  Hence there is
a collision in the `sha256` function, and therefore we can effectively find
a collision in the SHA256 compression function.  QED

## Avoiding collisions between merkleRoot and sha256
For the moment, let us assume there is no effective way to find a collision
in the SHA256 compression function.  By the Merkle--Damgård property of
SHA256, we cannot effectively find a collision within the sha256 function.
We have shown, by Theorem 2, that merkleRoot has the same Merkle--Damgård
property and hence we cannot effectively find a collision within the
merkleRoot function.  However, we may have collisions between the `sha256`
and the `merkleRoot` functions.  That is, we may be able to effectively
find values `s : BitString` and `t : Tree Word256` such that `sha256(s) =
merkleRoot(t)`.

While this may not be a problem for most applications, it turns out that we
can place restrictions on the format of tags to ensure that there are never
collisions between `sha256` and `merkleRoot`.  Define a safe tag of type
`Word256` to be a a value such that one the first 192 bits is set and the
9th last bit is cleared.

#### Theorem 6
Let `t : Tree Word256` be a tree in which every tag is a safe tag.  Suppose
we have some `s : BitString` such that `sha256(s) = merkleRoot(t)`.  Then
we can effectively find a collision in the SHA256 compression function.

Proof.
The last chunk of `sha256Padding(s)` is a padding chunk as defined by the
SHA256 standard.  If we look at the second half of this last chunk, then
according to the padding rules, it can never be the case that one of the
first 192 bits is set and the 9^th last bit is cleared.  Because `t` only
contains safe tags, the inputs to their last compression function in the
computation of `sha256(s)` and `merkleRoot(t)` must be different.
Therefore if `sha256(s) = merkleRoot(t)` then we must have encountered a
collision in the final compression function of the two computations.  QED

#### Remark 7
If we use safe tags we can consider a modified `merkleRoot'` function.

    merkleRoot' : Tree Word256 -> Word256
    merkleRoot' (Leaf (t))                :=
sha256Compress(sha256(ApplicationName),
sha256("") ⋅ t)
    merkleRoot' (Unary (t, child))        := sha256Compress(merkleRoot'(child),
sha256("") ⋅ t)
    merkleRoot' (Binary (t, left, right)) := sha256Compress(merkleRoot'(left),
merkleRoot'(right) ⋅ t)

If we use `merkleRoot'` we no longer require that tags uniquely define
their node types in order to ensure the Merkle--Damgård property.  Instead
we can rely on the safe tags to ensure that the use of `sha256(s)` will
never collide with `merkleRoot'(t)`, and therefore nodes of different types
will never collide with each other the `merkleRoot'` computation.  However,
I don't feel this is a particularly good approach, so I will not elaborate
on it further.

Unfortunately, there is no guarantee that the result of `hash256Tags` will
consist of only safe tags, so we cannot use it to avoid collisions between
`sha256` and `merkleRoot ⋅ hash256Tags`.  To remedy this we define an
alternative to `hash256Tags`.

    hash224Tag : BitString -> Word256
    hash224Tag (s) := 0xFFFF ⋅ sha224(s) ⋅ 0x0000

    hash224Tags : Tree BitString -> Tree Word256
    hash224Tags (t) := treeMap hash224Tag t

We see that by appropriately padding the result of `sha224` we guarantee
that `hash224Tag(s)` is a safe tag.

#### Theorem 8
Suppose `t_1` and `t_2` are two different `Tree BitString` values such that
any tag occurring in the two trees only occurs in one kind of node.  If
`merkleRoot(hash224Tags(t_1)) = merkleRoot(hash224Tags(t_2))`, then we can
effectively find either a collision in the SHA256 compression function, or
a collision in SHA224.

#### Remark 9
We can safely replace the `0xFFFF` prefix in `hash224Tag` with one where
the first two bits determine the kind of node in accordance with our
previously defined scheme.

#### Remark 10
Rather than using SHA224 we could use SHA256 and tweak it afterwards to set
the first bit and clear the 9th last bit. This comes at the cost of
effectively using a non-standard hash function.

#### Remark 11
Most of this development not depend specifically on SHA256.  It all works
similarly for SHA512 and other hash functions that compress in a similar
manner.  SHA256 is used as the primary example because one can find
hardware support for it on commodity hardware (for example, see the Intel
SHA extensions).

## Conclusion
We have defined a merkleRoot function for computing Merkle roots of trees
that include annotations per node.  The resulting function uses one SHA256
compression function call per node and supports arbitrary 256-bit
annotations per node.  Arbitrary annotations can be supported at the cost
of more hashing.  We have also shown that by using safe tags we can
additionally get a property that the `merkleRoot` will not effectively
collide with `sha256` function.

## Further Reading
The article [Characterizing Padding Rules of MD Hash Functions Preserving
Collision Security](https://eprint.iacr.org/2009/325.pdf) by Mridul Nandi
provides a nice overview of various options for padding rules.

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