On Thu, Jun 16, 2016 at 9:34 PM, Peter Todd <pete@petertodd.org> 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.