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

From: Peter Todd <[email-masked]>
Date: Fri, 17 Jun 2016 00:34:35 -0400

On Thu, Jun 16, 2016 at 02:07:26AM -0700, Bram Cohen wrote:
> On Wed, Jun 15, 2016 at 8:26 PM, Peter Todd <[email-masked]> wrote:
> Okay, clearly my assumptions about the parts of that post I didn't read
> carefully were way off. I'll have to look through it carefully to be able
> to make coherent apples to apples comparisons.

Thanks!

> > I'm worried that once there's real transaction fees everyone might stop
> > > consolidating dust and the set of unspent transactions might grow without
> > > bound as well, but that's a topic for another day.
> >
> > Ok, but then if you're concerned about that risk, why introduce a data
> > structure - the STXO set - that's _guaranteed_ to grow without bound?
> >
> 
> I'm not proposing STXO set commitments either. My point was that there
> should be incentives for collecting dust. That has nothing to do with this
> thread though and should be discussed separately (also I don't feel like
> discussing it because I don't have a good proposal).

Ah, yeah, I misunderstood you there; as expected absolutely no-one is proposing
STXO set commitments. :)

> > > The main differences to your patricia trie are the non-padding sha256 and
> > > that each level doesn't hash in a record of its depth and the usage of
> > > ONLY0 and ONLY1.
> >
> > I'm rather confused, as the above sounds nothing like what I've
> > implemented,
> > which only has leaf nodes, inner nodes, and the special empty node
> > singleton,
> > for both the MMR and merbinner trees.
> >
> 
> It's quite a bit like merbinner trees. I've basically taken the leaf nodes
> and smushed them into the inner nodes above them, thus saving a hashing
> operation and some memory. They're both binary radix trees.

Ah, I see what you mean now.

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.

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.

In my deterministic expressions work one of the ideas I've been tossing around
is rather than always using hash digests directly for when you need to commit
to some data, we could instead extend the idea of a digest to that of a
"commitment", where a commitment is simply some short, but variable-sized,
string that uniquely maps to a given set of data. Secondly, commitments do
*not* always guarantee that the original data can't be recovered from the
commitment itself.

By allowing commitments to be variable sized - say 0 to ~64 bytes - we get a
number of advantages:

1) Data shorter than the length of a digest (32 bytes) can be included in the
commitment itself, improving efficiency.

2) Data a little longer than a digest can have hashing delayed, to better fill
up blocks.

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.

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.

> > 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?
> 
> 
> My calculations are: a Bitcoin block contains about 2000 updates. The l2
> cache is about 256 kilobytes, and if an update is about 32 bytes times two
> for the parents, grandparents, etc. then an l2 cache can contain about 4000
> values. If the current utxo size is about 2000 * 4000 = 8,000,000 in size
> then about half the pages which contain a transaction will contain a second
> one. I think the utxo set is currently about an order of magnitude greater
> than that, so the number of such collisions will be fairly mall, hence my
> 'less than one but not a lot less' comment.

Your estimate of updates requiring 32 bytes of data is *way* off.

Each inner node updated on the path to a leaf node will itself require 32 bytes
of data to be fetched - the digest of the sibling. As of block 416,628, there
are 39,167,128 unspent txouts, giving us a tree about 25 levels deep.

So if I want to update a single leaf, I need to read:

    25 nodes * 32 bytes/node = 800 bytes

of data. Naively, that'd mean our 2,000 updates needs to read 1.6MB from RAM,
which is 6.4x bigger than the L2 cache - it's just not going to fit.

Taking into account the fact that this is a batched update improves things a
little bit. For a node at level i with random access patterns and N accesses
total our amortised cost is 1/(1 + N/2^i) Summing that over 2,000 leaf updates
and 25 levels gives us ~29,000 total updates, 0.9MB, which is still a lot
larger than L2 cache.

While this might fit in L3 cache - usually on the order of megabytes - this is
a rather optimistic scenario anyway: we're assuming no other cache pressure and
100% hit rate.

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.

> As for how it's indexed, at a crypto definition level it's just a binary
> radix tree. In terms of how it's indexed in memory, that involves some
> optimizations to avoid cache misses. Memory is allocated into blocks of
> about the size of an 12 cache (or maybe an l1 cache, it will require some
> testing and optimization). Blocks are either branch blocks, which keep
> everything in fixed positions, or leaf blocks, which contain fixed size
> entries for nodes plus indexes within the same leaf block of their
> children. Branch blocks can have many children which can be either branch
> blocks or leaf blocks, but typically are either all branch blocks or all
> leaf blocks. Branch blocks always have exactly one parent. Leaf blocks
> always have all their inputs come from a single branch block, but there can
> be multiple ones of those. When a branch block overflows it first tries to
> put stuff into the last leaf block it used, and if there's no more room it
> allocates a new one. It's fairly common for branches to have just a few
> leaf children, but they also could have a lot, depending on whether the
> base 2 log of the number of things currently in the set modulo the number
> levels in a branch is a small number.
> 
> Usually when an update is done it consists of first checking the
> appropriate output of the root block (it's jumped to directly to avoid
> unnecessary memory lookups. If there's nothing there the algorithm will
> walk back until it finds something.) That leads directly to (usually)
> another branch whose output is jumped to directly again. At Bitcoin utxo
> set sizes that will usually lead to a leaf block, which is then walked down
> manually to find the actual terminal node, which is then updated, and the
> parent, grandparent, etc. is then marked invalid until something which was
> already marked invalid is hit, and it exits. Calculation of hash values is
> done lazily.

I think it's safe to say that given our working set is significantly larger
than the L2/L3 cache available, none of the above optimizations are likely to
help much. Better to just keep the codebase simple and use standard techniques.

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