Subject: Re: [bitcoin-dev] Merkle trees and mountain ranges

From: Bram Cohen <[email-masked]>
Date: Fri, 17 Jun 2016 19:43:47 -0700

On Thu, Jun 16, 2016 at 9:34 PM, Peter Todd <[email-masked]> wrote:

> So above you said that in merbinner trees each node "hash[es] in a record
> of
> its depth" That's actually incorrect: each node commits to the prefix that
> all
> keys below that level start with, not just the depth.

I considered a similar trick at the implementation rather than the
definition level: A node doesn't have to store the prefix which is implicit
in its position. That would create a fair number of headaches though,
because I'm using fixed size stuff in important ways, and it could at most
save about 10% of memory, so it goes into the 'maybe later' bucket.

>
> This means that in merbinner trees, cases where multiple keys share parts
> of
> the same prefix are handled efficiently, without introducing extra levels
> unnecessarily; there's no need for the ONLY0/1 nodes as the children of an
> inner node will always be on different sides.
>
> When keys are randomly distributed, this isn't a big deal; OTOH against
> attackers who are choosing keys, e.g. by grinding hashes, merbinner trees
> always have maximum depths in proportion to log2(n) of the actual number of
> items in the tree. Grinding is particularly annoying to deal with due to
> the
> birthday attack: creating a ground prefix 64 bits long only takes 32 bits
> worth
> of work.
>

Yes an attacker can force the tree to be deeper in places, but it's
mitigated in several ways: (1) The way I'm using memory it won't cause a
whole new block to be allocated, it will just force log(attack strength) -
log(n) nodes to be used (2) logarithmic growth being what it is that isn't
such a big amount (3) With the special casing of TERMBOTH an attacker needs
three things with the same prefix to pull off an attack rather than two,
which is quite a bit harder to pull off.

That said, it wouldn't be all that hard to change how the hashing function
works to do a single hash for a whole series of ONLY in a row instead of a
new one at every level, which would make the attacker only able to force
extra memory usage instead of extra CPU, but this is a slightly annoying
thing to write to stop a fairly lame attack, so I'm at least not doing it
for my initial implementation. I could likely be convinced that it's worth
doing before an actual release though. There's another implementation trick
to do the same thing for memory usage, which is much more in the 'do later'
category because it doesn't involve changing the format and hence it can be
put off.

> In particular, case #2 handles your leaf node optimizations generically,
> without special cases and additional complexity. It'd also be a better way
> to
> do the ONLY0/1 cases, as if the "nothing on this side" symbol is a single
> byte,
> each additional colliding level would simply extend the commitment without
> hashing. In short, you'd have nearly the same level of optimization even
> if at
> the cryptography level your tree consists of only leaves, inner nodes, and
> nil.
>

I'm taking pains to make all the hashing be of fixed-size things, so that a
non-padding variant of a secure hashing algorithm can be used. The chains
of ONLY thing above would force a special exception to that, which can be
done but is annoying. Making things smaller than a single block (64 bytes)
won't speed up hashing time, and making things a single byte longer than
that doubles it.

> Another advantage of variable sized commitments is that it can help make
> clear
> to users when it's possible to brute force the message behind the
> commitment.
> For instance, digest from a hashed four byte integer can be trivially
> reversed
> by just trying all combinations. Equally, if that integer is concatenated
> with
> a 32 byte digest that the attacker knows, the value of the integer can be
> brute
> forced.
>

I'm hashing all strings before inserting to get them to be a fixed size and
avoid a few different attacks. In Bitcoin most of the strings added are
longer than that so it's a form of compression. A custom hash function
could be used which 'hashes' very short strings by repeating them verbatim
could be used, but seems like not such a hot idea. I'm making extensive use
of things being fixed size everywhere, which improves performance in a lot
of ways.

> > > Technically even a patricia trie utxo commitment can have sub-1 cache
> > > > misses per update if some of the updates in a single block are close
> to
> > > > each other in memory. I think I can get practical Bitcoin updates
> down
> > > to a
> > > > little bit less than one l2 cache miss per update, but not a lot
> less.
> > >
> > > I'm very confused as to why you think that's possible. When you say
> > > "practical
> > > Bitcoin updates", what exactly is the data structure you're proposing
> to
> > > update? How is it indexed?
>

I'll re-answer this because I did a terrible job before. The entire data
structure consists of nodes which contain a metadata byte (TERM0, ONLY1,
etc.) followed by fixes size secure hashes, and (in some cases) pointers to
where the children are. The secure hashes in parent nodes are found by
hashing their children verbatim (or the stored string in the case of a
TERM). This is very conventional. All of the cleverness is in where in
memory these nodes are stored so that tracing down the tree causes very few
cache misses.

(The alternate approach is to have each node store its own hash rather than
that be stored by the parent. That approach means that when you're
recalculating you have to look up siblings which doubles the number of
cache misses. Not such a hot idea.)

At the root there's a branch block. It consists of all nodes up to some
fixed depth - let's say 12 - with that depth set so that it roughly fits
within a single memory page. Branch blocks are arranged with the nodes in
fixed position defined by the prefix they correspond to, and the terminals
have outpointers to other blocks. Because they're all clustered together, a
lookup or update will only require a single

Below the root block are other branch blocks. Each of them has a fixed 12
bit prefix it is responsible for. When doing a lookup a second cache miss
will be hit for levels 13-24, because those are all clustered in the same
branch block.

Below the second level of root block (at Bitcoin utxo set scale - this
varies based on how much is stored) there are leaf blocks. A leaf block
consists of nodes with outpointers to its own children which must be within
the same leaf block. All entry points into a leaf block are from the same
branch block, and the leaf block has no out pointers to other blocks. When
a leaf block overflows the entry point into it which overflowed is moved
into the active leaf for that branch, and if that's full a new one is
allocated. There's some subtlety to exactly how this is done, but I've
gotten everything down to simple expedient tricks with straightforward
implementations. The thing which matters for now is that there's only a
single cache miss for each leaf node, because they also fit in a page.

So at Bitcoin scale there will probably only be 3 cache misses for a
lookup, and that's a worst case scenario. The first one is probably always
warm, bringing it down to 2, and if you do a bunch in sorted order they'll
probably hit the same second level branches repeatedly bringing it down to
1, and might even average less than that if there are enough that the leaf
block has multiple things being accessed.

(These same tricks can be applied to merbinner tree implementation as well,
although that would be a bit less parsimonious with memory, by a small
constant factor.)

> Anyway hashing is pretty slow. The very fast BLAKE2 is about 3 cycles/byte
> (SHA256 is about 15 cycles/byte) so hashing that same data would take
> around
> 200 cycles, and probably quite a bit more in practice due to overheads
> from our
> short message lengths; fetching a cache line from DRAM only takes about
> 1,000
> cycles. I'd guess that once other overheads are taken into account, even
> if you
> could eliminate L2/L3 cache-misses it wouldn't be much of an improvement.
>

Those numbers actually back up my claims about performance. If you're doing
a single update and recalculating the root afterwards, then the amount of
rehashing to be done is about 30 levels deep times 64 bytes per thing
hashed times 15 cycles per byte then it's about 28,800 cycles of hashing.
If you have a naive radix tree implementation which hits a cache miss at
every level then that's 30,000 cycles, which is about half the performance
time, certainly worth optimizing. If instead of sha256 you use blake2
(Which sounds like a very good idea!) then hashing for an update will be
about 5760 cycles and performance will be completely dominated by cache
misses. If a more cache coherent implementation is used, then the cost of
cache misses will be 3000 cycles, which will be a non-factor with sha256
and a significant but not dominating one with blake2.

It's reasonable to interpret those numbers as saying that blake2 and cache
coherent implementation are both clearly worth it (probably necessary for
real adoption) and that an amortized binary radix tree is tempting but not
worth it.