A Method for Computing Merkle Roots of Annotated Binary Trees

A Method for Computing Merkle Roots of Annotated Binary Trees
2017-05-22 14:05 ` Peter Todd
                   ` (2 more replies)
  
From: Russell O'Connor @ 2017-05-22  7:05 UTC (permalink / raw)
  To: Bitcoin Protocol Discussion

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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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 permalink raw reply	[|nested]
2017-05-22 14:05 ` 
                   ` 
  
From: Russell O'Connor @ 2017-05-22  7:05 UTC (permalink / raw)
  To: Bitcoin Protocol Discussion

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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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2017-05-22 14:05 ` Peter Todd

  2017-05-27 17:41     ` Peter Todd
  
From: Russell O'Connor @ 2017-05-22 22:32 UTC (permalink / raw)
  To: Peter Todd;  Bitcoin Protocol Discussion

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On May 22, 2017 23:05, "Peter Todd" <pete@petertodd•org> wrote:

On Mon, May 22, 2017 at 03:05:49AM -0400, Russell O'Connor via bitcoin-dev
wrote:
>     MerkleRoot := SHA256(SHA256(LeftRoot ⋅ RightRoot))
>     sha256Compress : Word256 × Word512 -> Word256

To be clear, what math operations do you mean by "⋅" and "×"?


By "⋅", I usually mean concatenation (though I also use it for function
composition in one instance).   By "×", I mean the Cartesian product.

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2017-05-22  7:05 [bitcoin-dev] A Method for Computing Merkle Roots of Annotated Binary Trees Russell O'Connor
  2017-05-22 14:05 ` Peter Todd

  2017-05-28  8:26 ` Peter Todd
  
From: Bram Cohen @ 2017-05-23  6:06 UTC (permalink / raw)
  To: Russell O'Connor;  Bitcoin Protocol Discussion

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On Mon, May 22, 2017 at 12:05 AM, Russell O'Connor via bitcoin-dev <
bitcoin-dev@lists•linuxfoundation.org> wrote:

> 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.
>

Ya know, when you're building a Merkle Trie that's exactly the primitive
you need.

In my own construction the assumption is that the values are already hashed
down to 256 bits when they're passed in, and the tags (which are currently
done by sacrificing bits instead of using tags, that needs to be fixed)
include three states for either side: empty, unary, or middle. Three of
those possibilities are unreachable (empty/empty, empty/unary, unary/empty)
so there are 6 possible tags needed. This approach essentially skips doing
the unary hashes, a further performance improvement. There doesn't appear
to be any downside in leveraging this trick as long as tags are cheap.

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2017-05-22 22:32   ` Russell O'Connor

       [not found]       ` <CAMZUoKkS8azx7Gooo3D+H_gdGdTNiNtwwNVbvU0u7HzOfdUSBg@mail.gmail.com>
  
From: Peter Todd @ 2017-05-27 17:41 UTC (permalink / raw)
  To: Russell O'Connor;  Bitcoin Protocol Discussion

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On Mon, May 22, 2017 at 06:32:38PM -0400, Russell O'Connor wrote:
> On May 22, 2017 23:05, "Peter Todd" <pete@petertodd•org> wrote:
> 
> On Mon, May 22, 2017 at 03:05:49AM -0400, Russell O'Connor via bitcoin-dev
> wrote:
> >     MerkleRoot := SHA256(SHA256(LeftRoot ⋅ RightRoot))
> >     sha256Compress : Word256 × Word512 -> Word256
> 
> To be clear, what math operations do you mean by "⋅" and "×"?
> 
> 
> By "⋅", I usually mean concatenation (though I also use it for function
> composition in one instance).   By "×", I mean the Cartesian product.

Cartesian product can mean a lot of things.

What specifically do you mean by "cartesian product" here?

-- 
https://petertodd.org 'peter'[:-1]@petertodd.org

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[not found]       ` <CAMZUoKkS8azx7Gooo3D+H_gdGdTNiNtwwNVbvU0u7HzOfdUSBg@mail.gmail.com>

  
From: Russell O'Connor @ 2017-05-27 22:07 UTC (permalink / raw)
  To: Peter Todd, Bitcoin Protocol Discussion

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On May 28, 2017 06:09, "Russell O'Connor" <roconnor@blockstream•io> wrote:



On May 28, 2017 03:16, "Peter Todd" <pete@petertodd•org> wrote:

On Mon, May 22, 2017 at 06:32:38PM -0400, Russell O'Connor wrote:
> On May 22, 2017 23:05, "Peter Todd" <pete@petertodd•org> wrote:
>
> On Mon, May 22, 2017 at 03:05:49AM -0400, Russell O'Connor via bitcoin-dev
> wrote:
> >     MerkleRoot := SHA256(SHA256(LeftRoot ⋅ RightRoot))
> >     sha256Compress : Word256 × Word512 -> Word256
>
> To be clear, what math operations do you mean by "⋅" and "×"?
>
>
> By "⋅", I usually mean concatenation (though I also use it for function
> composition in one instance).   By "×", I mean the Cartesian product.

Cartesian product can mean a lot of things.

What specifically do you mean by "cartesian product" here?


Oops, I forgot to reply all.  Below is my reply.

Given two types A and B, then A × B is the type of pairs of A and B in the
sense of type theory as used in Standard ML or Haskell or other typed
languages.


To follow up, by "sha256Compress : Word256 × Word512 -> Word256" I mean
that sha256Compress is a function that takes two arguments, the first being
a 256 bit word and the second being a 512 bit word, and returns a 256 bit
word (or equivalently sha256Compress is a function that takes a pair as
input, the first component being a 256 bit word and the second component
being a 512 bit word, and returns a 256 bit word).

sha256Compress is meant to be the compression function defined by the
SHA256 standard, though nothing here depends on anything more that its type
signature.

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2017-05-22  7:05 [bitcoin-dev] A Method for Computing Merkle Roots of Annotated Binary Trees Russell O'Connor
  2017-05-22 14:05 ` Peter Todd
  2017-05-23  6:06 ` Bram Cohen

  2017-05-29 14:55   ` Russell O'Connor
  
From: Peter Todd @ 2017-05-28  8:26 UTC (permalink / raw)
  To: Russell O'Connor;  Bitcoin Protocol Discussion

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On Mon, May 22, 2017 at 03:05:49AM -0400, Russell O'Connor via bitcoin-dev wrote:
> 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`.

This doesn't hold true in the case of pruned trees, as for the pruning to be
useful, you don't know what produced the left merkleRoot, and thus you can't
guarantee it is in fact a midstate of a genuine SHA256 hash.

-- 
https://petertodd.org 'peter'[:-1]@petertodd.org

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2017-05-28  8:26 ` Peter Todd

  2017-05-29 16:10     ` Peter Todd
  2017-06-27  4:13     ` Peter Todd
  
From: Russell O'Connor @ 2017-05-29 14:55 UTC (permalink / raw)
  To: Peter Todd;  Bitcoin Protocol Discussion

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On Sun, May 28, 2017 at 4:26 AM, Peter Todd <pete@petertodd•org> wrote:

> On Mon, May 22, 2017 at 03:05:49AM -0400, Russell O'Connor via bitcoin-dev
> wrote:
> > 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`.
>
> This doesn't hold true in the case of pruned trees, as for the pruning to
> be
> useful, you don't know what produced the left merkleRoot, and thus you
> can't
> guarantee it is in fact a midstate of a genuine SHA256 hash.
>

Thanks for the review Peter.  This does seem like a serious issue that I
hadn't considered yet.  As far as I understand, we have no reason to think
that the SHA-256 compression function will be secure with chosen initial
values.

Some of this proposal can be salvaged, I think, by putting the hash of the
tags into Sha256Compress's first argument:

    merkleRoot : Tree BitString -> Word256
    merkleRoot (Leaf (t))                := sha256Compress(sha256(t),
0b0^512)
    merkleRoot (Unary (t, child))        := sha256Compress(sha256(t),
merkleRoot(child) ⋅ 0b0^256)
    merkleRoot (Binary (t, left, right)) := sha256Compress(sha256(t),
merkleRoot(left) ⋅ merkleRoot(right))

The Merkle--Damgård property will still hold under the same conditions
about tags determining the type of node (Leaf, Unary, or Binary) while
avoiding the need for SHA256's padding.  If you have a small number of
tags, then you can pre-compute their hashes so that you only end up with
one call to SHA256 compress per node. If you have tags with a large amount
of data, you were going to be hashing them anyways.

Unfortunately we lose the ability to cleverly avoid collisions between the
Sha256 and MerkleRoot functions by using safe tags.

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2017-05-29 14:55   ` Russell O'Connor

  2017-06-01 15:10       ` Russell O'Connor
  2017-06-27  4:13     ` Peter Todd
  
From: Peter Todd @ 2017-05-29 16:10 UTC (permalink / raw)
  To: Russell O'Connor;  Bitcoin Protocol Discussion

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On Mon, May 29, 2017 at 10:55:37AM -0400, Russell O'Connor wrote:
> > This doesn't hold true in the case of pruned trees, as for the pruning to
> > be
> > useful, you don't know what produced the left merkleRoot, and thus you
> > can't
> > guarantee it is in fact a midstate of a genuine SHA256 hash.
> >
> 
> Thanks for the review Peter.  This does seem like a serious issue that I
> hadn't considered yet.  As far as I understand, we have no reason to think
> that the SHA-256 compression function will be secure with chosen initial
> values.

Well, it's an easy thing to forget, unless like me, you've written a general
purpose library to work with pruned data. :) (actually, soon to be two of
them!)

I also ran into the midstate issue with my OpenTimestamps protocol, as I was
looking into whether or not it was safe to have a SHA256 midstate commitment
operation, and couldn't find clear evidence that it was.

> Some of this proposal can be salvaged, I think, by putting the hash of the
> tags into Sha256Compress's first argument:
> 
>     merkleRoot : Tree BitString -> Word256
>     merkleRoot (Leaf (t))                := sha256Compress(sha256(t),
> 0b0^512)
>     merkleRoot (Unary (t, child))        := sha256Compress(sha256(t),
> merkleRoot(child) ⋅ 0b0^256)
>     merkleRoot (Binary (t, left, right)) := sha256Compress(sha256(t),
> merkleRoot(left) ⋅ merkleRoot(right))
> 
> The Merkle--Damgård property will still hold under the same conditions
> about tags determining the type of node (Leaf, Unary, or Binary) while
> avoiding the need for SHA256's padding.  If you have a small number of
> tags, then you can pre-compute their hashes so that you only end up with
> one call to SHA256 compress per node. If you have tags with a large amount
> of data, you were going to be hashing them anyways.

Notice how what you're proposing here is almost the same thing as using SHA256
directly, modulo the fact that you skip the final block containing the message
length.

Similarly, you don't need to compute sha256(t) - you can just as easily compute
the midstate sha256Compress(IV, t), and cache that midstate if you can reuse
tags. Again, the only difference is the last block.

> Unfortunately we lose the ability to cleverly avoid collisions between the
> Sha256 and MerkleRoot functions by using safe tags.

I think a better question to ask is why you want that property in the first
place?

My earlier python-proofmarshal(1) library had a scheme of per-use-case tags,
but I eventually realised that depending on tags being unique is a footgun. For
example, it's easy to see how two different systems could end up using the same
tag due to designers forgetting to create new tags while copying and pasting
old code. Similarly, if two such systems have to be integrated, you'll end up
with tags getting reused for two different purposes.

Now, if you design a system where that doesn't matter, then by extension it'll
also be true that collisions between the sha256 and merkleroot functions don't
matter either. And that system will be more robust to design mistakes, as tags
only need to be unique "locally" to distinguish between different sub-types in
a sum type (enum).


FWIW what I've done with my newer (and as yet unpublished) rust-proofmarshal
work is commitments are only valid for a specific type. Secondly, I use
blake2b, whose compression function processes up to 128 bytes of message on
each invocation.  That's large enough for four 32 byte hashes, which is by
itself more than sufficient for a summed merkle tree with three 32 byte hashes
(left right and sum) and a per-node-type tag.

Blake2b's documentations don't make it clear if it's resistant to collision if
the adversary can control the salt or personalization strings, so I don't
bother using them - the large block size by itself is enough to fit almost any
use-case into a single block, and it hashes blocks significantly faster than
SHA256. This also has the advantage that the actual primitive I'm using is 100%
standard blake2b, an aid to debugging and development.

1) https://github.com/proofchains/python-proofmarshal

-- 
https://petertodd.org 'peter'[:-1]@petertodd.org

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2017-05-29 16:10     ` Peter Todd

  
From: Russell O'Connor @ 2017-06-01 15:10 UTC (permalink / raw)
  To: Peter Todd;  Bitcoin Protocol Discussion

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On Mon, May 29, 2017 at 12:10 PM, Peter Todd <pete@petertodd•org> wrote:

> On Mon, May 29, 2017 at 10:55:37AM -0400, Russell O'Connor wrote:
> > Some of this proposal can be salvaged, I think, by putting the hash of
> the
> > tags into Sha256Compress's first argument:
> >
> >     merkleRoot : Tree BitString -> Word256
> >     merkleRoot (Leaf (t))                := sha256Compress(sha256(t),
> > 0b0^512)
> >     merkleRoot (Unary (t, child))        := sha256Compress(sha256(t),
> > merkleRoot(child) ⋅ 0b0^256)
> >     merkleRoot (Binary (t, left, right)) := sha256Compress(sha256(t),
> > merkleRoot(left) ⋅ merkleRoot(right))
> >
> > The Merkle--Damgård property will still hold under the same conditions
> > about tags determining the type of node (Leaf, Unary, or Binary) while
> > avoiding the need for SHA256's padding.  If you have a small number of
> > tags, then you can pre-compute their hashes so that you only end up with
> > one call to SHA256 compress per node. If you have tags with a large
> amount
> > of data, you were going to be hashing them anyways.
>
> Notice how what you're proposing here is almost the same thing as using
> SHA256
> directly, modulo the fact that you skip the final block containing the
> message
> length.
>
> Similarly, you don't need to compute sha256(t) - you can just as easily
> compute
> the midstate sha256Compress(IV, t), and cache that midstate if you can
> reuse
> tags. Again, the only difference is the last block.
>

Yes, either way would be fine.


> > Unfortunately we lose the ability to cleverly avoid collisions between
> the
> > Sha256 and MerkleRoot functions by using safe tags.
>
> I think a better question to ask is why you want that property in the first
> place?
>
> My earlier python-proofmarshal(1) library had a scheme of per-use-case
> tags,
> but I eventually realised that depending on tags being unique is a
> footgun. For
> example, it's easy to see how two different systems could end up using the
> same
> tag due to designers forgetting to create new tags while copying and
> pasting
> old code. Similarly, if two such systems have to be integrated, you'll end
> up
> with tags getting reused for two different purposes.
>
> Now, if you design a system where that doesn't matter, then by extension
> it'll
> also be true that collisions between the sha256 and merkleroot functions
> don't
> matter either. And that system will be more robust to design mistakes, as
> tags
> only need to be unique "locally" to distinguish between different
> sub-types in
> a sum type (enum).
>

I was looking to add the property mostly because it was free to do with my
original design when the set of tags was small and could make some
applications more robust and/or easier to debug.

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2017-05-29 14:55   ` Russell O'Connor
  2017-05-29 16:10     ` Peter Todd

  
From: Peter Todd @ 2017-06-27  4:13 UTC (permalink / raw)
  To: Russell O'Connor;  Bitcoin Protocol Discussion

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On Mon, May 29, 2017 at 10:55:37AM -0400, Russell O'Connor wrote:
> > This doesn't hold true in the case of pruned trees, as for the pruning to
> > be
> > useful, you don't know what produced the left merkleRoot, and thus you
> > can't
> > guarantee it is in fact a midstate of a genuine SHA256 hash.
> >
> 
> Thanks for the review Peter.  This does seem like a serious issue that I
> hadn't considered yet.  As far as I understand, we have no reason to think
> that the SHA-256 compression function will be secure with chosen initial
> values.

Relevant: fixed points can be found for the SHA256 compression function, if the
attacker can control the IV:

https://crypto.stackexchange.com/questions/48580/fixed-point-of-the-sha-256-compression-function

-- 
https://petertodd.org 'peter'[:-1]@petertodd.org

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