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12:18 < maaku> empirical observation: the maximum number of branch hashes required to prove membership of any subset of the leaves of a tree of height N is (N + 1) / 2
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12:18 < maaku> does anyone know how to prove this?
12:19 < kanzure> "tree of height N" what does that mean
12:19 < maaku> A tree for which the maximum depth of any leaf is N
12:20 < maaku> wait n/m I may have messed something up
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12:22 < esotericnonsense> maaku: eh? it has to be at least N.
12:22 < kanzure> merkle tree inclusion proof size grows with depth but it should be constant growth
12:22 < maaku> ookaay yeah that seemed too incredible to be true, hence my confusion. and it wasn't true as I was outputing the wrong value for N
12:23 < maaku> so what was being observed is that to prove k-of-n requires max (n+1)/2 inner hashes for any k
12:23 < maaku> which sounds more right, but I'm still interested in proving that
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12:24 < maaku> (it's true up to k-of-26 by exhaustive testing)
12:24 < esotericnonsense> what is 'k' in this context
12:24 < kanzure> are these binary merkle trees
12:24 < maaku> kanzure: yes
12:25 < kanzure> (2n + c) where n = depth * hash length, and c = size of whatever content you're checking at the very end, right?
12:25 < sipa> maaku: huh?
12:25 < sipa> i think i'm misunderstanding the statement
12:25 < maaku> esotericnonsense: for k=3,n=5, 3-of-5, that means proving 3 elements out of a balanced binary tree of 5 elements
12:25 < esotericnonsense> right
12:26 < sipa> if you have a balanced 1024-element tree, so N = 10, proving the inclusion of the first and last element requires 19 hashes, I believe
12:27 < esotericnonsense> yeah, i don't understand how this (n+1)/2 can come about unless you're excluding the hashes of the elements themselves
12:27 < sipa> even if you exclude them
12:27 < esotericnonsense> choose an element at the 'bottom'
12:28 < maaku> sipa: example: "n=8 max=4 1 14 64 112 64"" <-- for proving inclusion of various combinations of 8 keys, there is 1 combination that takes 0 inner hashes, 14 that take 1 hash, 64 that take 2 hashes, 112 that take 3 hashes, and 64 that take 4 hashes
12:28 < maaku> observation max=(n+1)/2
12:28 < sipa> what are those numbers?
12:29 < maaku> https://0bin.net/paste/5hdjPpSt6GxVX1UW#ptfohomsUZ66nUzGBSstDXmh4V8Fbll2Q+vRNrVFhi2
12:29 < sipa> can you explain what's wrong with my example?
12:29 < sipa> 19 > (10+1)/2
12:29 < esotericnonsense> even just the 1 example breaks it
12:30 < maaku> sipa: you're talking about what I mistakingly wrote first, not what I corrected it to
12:30 < sipa> maaku: i must be misunderstanding your correction then
12:30 < sipa> oh!
12:30 < sipa> N is number of elements, not tree depth
12:31 < maaku> the observation would be that for a 1024 element tree, the maximum number of hashes needed to prove inclusion of some subset of those hashes would be (1024 + 1) / 2 = 512 hashes
12:31 < esotericnonsense> doh :P
12:32 < sipa> interesting!
12:32 < esotericnonsense> maaku: is that excluding the hashes themselves
12:32 < sipa> so he's just talking about the proof size, not the amount of computation done to verify a proof
12:32 < maaku> yes, it requires k+max hashes, k being the number of elements you are proving
12:33 < maaku> to rebuild the root
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12:33 < esotericnonsense> unless i'm missing something the worst case seems trivial
12:34 < esotericnonsense> you have your bottom level which has 1024 elements, you are given one from each branch at the bottom
12:34 < esotericnonsense> so you need the other 512 (each branch at the bottom level you don't have)
12:34 < maaku> sipa: those numbers above are a histogram of the various proof sizes for all thresholds over n keys
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12:36 < maaku> I was trying to see how many keys could be supported before proofs exceed push limitations of script
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12:37 < maaku> and if that relationship holds, the answer is 31, maybe 30 because of other info in the proof
12:37 < esotericnonsense> with 1023 your worst case is to have one from all the even sets (511) missing plus the standalone one (your subset contains the other 511)
12:38 < esotericnonsense> sorry if my terminology is crap here, just visualizing it
12:38 < kanzure> 30 is ~2x over p2sh ya?
12:38 < esotericnonsense> but once you've covered the even and odd case you can generalize from there
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12:38 < kanzure> oh it's only for a single push
12:39 < sipa> maaku: none of these problems exist when the merkle proof is not part of the script...
12:39 < maaku> sipa: which would be a good reason to figure out the extent of these problems!
12:40 < sipa> fair enough
12:41 < maaku> the code, fwiw : https://0bin.net/paste/lqO89q8Sjn4iJvTZ#uYl2sQJ-tocTAt0NYgN76uOX2hCZ6Q+QB2X/H25gXOT
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12:49 < maaku> esotericnonsense: yes that makes sense. it's not a rigorous proof but it primes my intuition
12:50  * esotericnonsense is an (ex) physicist
12:50 < esotericnonsense> we don't really do proof
12:50 < esotericnonsense> :D
12:50 < esotericnonsense> (I do often regret not studying mathematics)
12:52 < maaku> actually I guess you can show that it is as bad or worse to have one of two children missing than to elide the branch entirely
12:52 < maaku> then inductively the worst case would be as you describe, one of each pair at the bottom of the tree, plus one for the hanging-off odd element
12:52 < maaku> cool, thanks :)
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12:57 < sipa> maaku: it's a very interesting property to know
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12:58 < gmaxwell> maaku: thats what I previously believed the maximum was, I believe the worst case is where you have every other hash.
12:58 < gmaxwell> and then have to provide its sibling hash.
13:01 < gmaxwell> There was a question I ponded before related to this where, say you have a N element set, now you replicate it so it's 2N and permute the replica. And add a constraint that the verifier can traverse into one replica or another. By choosing the permutation correctly you can select one where the worse case in one tree is efficient in the other.  So you've increased your minimum proof size by one, b
13:01 < gmaxwell> ut I think you halve the maximum proof size.
13:01 < gmaxwell> (with the permutation being all even elements then all odd elements)
13:08 < esotericnonsense> i don't think you halve it
13:10 < esotericnonsense> hard for me to write down what I believe to be a counterexample
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13:12 < esotericnonsense> nah, it doesn't work. scratch that.
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13:23 < esotericnonsense> actually i see now.
13:24 < esotericnonsense> it seems to me that you go from 1 + (N+1)/2 to 2 + (N+1)/4
13:24 < esotericnonsense> where 1/2 correspond to the root hashes
13:25 < esotericnonsense> can you continue to make tradeoffs with other permutations and push the min/max closer together?
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13:30 < esotericnonsense> ah, you don't have to include the second root hash in the worst case because it can be derived.
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13:30 < esotericnonsense> (as the rule of using even/odd is defined).
13:35 < gmaxwell> To halve the worst case again, you add two more copies of N, one does the even odd flip on the second to last level, and one does it on both that and the last level.
13:35 < gmaxwell> at the expense of adding one more level to the best case.
13:35 < gmaxwell> I believe.
13:35 < esotericnonsense> very neat.
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13:45 < gmaxwell> maaku: this also is a reason that the scheme should not cap the number you pick but instead cap the maximum hashes, since with the above encoding you can make it so the maximum path is shorter than you'd expect.
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14:06 < maaku> Right, push limits cap you to a 4-of-32 or 28-of-32, unless you do some trick as you say. But to do 26-of-32 you could prove 28-of-32 and then drop two of them.
14:06 < maaku> Lots of little tricks to work around the edges.
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14:13 < esotericnonsense> the first genuinely valid use case for my silly-large monitor has been to spreadsheet gmaxwell's ' with an N=16
14:13 < esotericnonsense> *gmaxwell's 'halve it again' algorithm.
14:14 < esotericnonsense> can't find the counterexample. :)
14:16 < esotericnonsense> (i tried to do it on paper and ran out of space.)
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