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07:55 < kanzure> "The discrete logarithm problem over prime fields can be transformed to a linear multivariable Chinese remainder theorem" https://arxiv.org/abs/1608.07032
07:55 < kanzure> "We show that the classical discrete logarithm problem over prime fields can be reduced to that of solving a system of linear modular equations."
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08:17 < Taek> amiller, andytoshi, I think I'm missing information. So you have a bunch of SPV blocks that demonstrate X much work was done, but how to you prove that it was all done on a linked-list of blocks?
08:17 < Taek> What's to stop me from grabbing blocks from the real chain to show that my attack chain has a bunch of work?
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08:18 < andytoshi> Taek: sorry, there is missing context. suppose every block commits to every previous block in a merkle tree, rather than having just a prevblockhash link
08:18 < Taek> ah
08:18 < andytoshi> then i can give you a chain from tip to genesis where i've dropped a bunch of blocks, and you can use the extra backlinks to see that it follows all the way through
08:18 < andytoshi> there is a data structure called a "skiplist" that has a paper about it, even though pretty-much all the intuition is in its name :)
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08:20 < amiller> there's still more context needed, it's pretty non-trivial how to compare two proof of work samples in a way that guarantees forks don't cause a problem, this isn't addressed in the sidechains paper
08:21 < amiller> i think that the algorithm in this paper does the trick but that it wasn't fleshed out until here http://fc16.ifca.ai/bitcoin/papers/KLS16.pdf
08:21 < andytoshi> yeah, the sidechains paper was using them basically as a witness where uniqueness was not important
08:21 < andytoshi> oh, nice
08:23 < andytoshi> if you were actually trying to do pruning, i think algorithm in the paper can run into trouble ... imagine i give Taek  blocks 1, 2, 4 (with a skiplink from 4 to 2), then you make block 5 which is supposed to commit to 1, 2, 3, 4
08:23 < andytoshi> but if the commitment to 3 is missing/bad Taek can't tell
08:24 < andytoshi> and you can cause forks this way if people have different pruned lists
08:25 < andytoshi> i'll read this paper, thank you, maybe it solves it. (i think i solved it myself but in an adhoc way)
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08:34 < andytoshi> ah, ok, so this is really about SPV proofs and not about chain selection
08:34 < kanzure> a musical performance by djb and tanja lange https://www.youtube.com/watch?v=t37sXxA2pmw
08:40 < andytoshi> amiller: oh, you meant something different from what i was thinking by "forks don't cause a problem", i have been thinking about actually pruning blockchains with compact SPV and have been worried about the compact SPV itself causing forks where people don't agree on whether new blocks extend the chain
08:40 < andytoshi> which i think is not something anybody has really looked at
08:41 < amiller> right, it doesn't help with like SPV mining.... that's still unsafe in the same way
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08:44 < andytoshi> well, what i'm thinking is, imagine you have a 'canonical compact chain', where every block is required to skip (a) to the previous block; (b) as far back as it can in the already-pruned chain. (you don't know (b) til you've mined it so you've gotta commit to a lot of blocks, but only those two will matter in the end)
08:44 < andytoshi> then you explicitly commit to the block indices in the already-pruned chain, so there's consensus on that
08:47 < andytoshi> then validators need to check only two merkle proofs, and the same ones for every verifier. if they've got the compact chain up to the point of the block they're validating, they know what indices need to be committed to
08:48 < andytoshi> there is the problem that compact verifiers cannot check the link to the previous block, but this is a much smaller consensus risk
08:49 < andytoshi> so i'm thinking of like, a mimblewimble chain where everyone has the last k blocks (for k = 10000 or some security parameter) and just a compact spv proof from that back to the genesis
08:51 < andytoshi> the compact spv proof could be missing backlinks, and this would mean the chain wouldn't have been verified by anyone doing full verification back then, but this is entirely irrelevant to the chain's operation _now_
09:10 < maaku> kanzure: is there any interesting ramifications to that?
09:11 < kanzure> andytoshi: hm? why not store and commit to to old spv proofs? storage by unrelated third parties, store hash in chain, later procure it as evidence that things seemed to verify in the past, ya?
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09:11 < andytoshi> kanzure: sure, anyone who wanted to check on that could surely find the data
09:12 < kanzure> maaku: well, it means we need to brush up on piano and outwit djb with our own musical team
09:13 < maaku> kanzure: I mean the chinese remainder theorem thing you posted earlier
09:13 < maaku> the paper is a bit sparse on ramifications
09:13 < kanzure> maaku: oh. maybe el gamal? not sure. i am not a cryptographer :).
09:13 < maaku> and I don't know much about CRT
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09:43 < andytoshi> maaku: trying to understand the asymptotics of the magic on page 3
09:44 < andytoshi> i can tell you they don't understand the ramifications themselves, they say "multivariate CRT is hard lol" but typing "multivariate chinese remainder theorem" into google gets you http://www.math.harvard.edu/~knill/preprints/linear.pdf which points out a way to solve it in many circumstances
09:46 < andytoshi> maaku: like, they end with this example "compute the DL of 4 with respect to 2 mod 11" (btw this is 2, 2^2 = 4 :)). they go through their mechanations to get "beta + n = 6 mod 11", "beta + 2n = 3 mod 5". by looking for solutions on the curve (beta, n) = (2t, t) this is {3t = 6 mod 11, 4t = 3 mod 5} or {t = 8 mod 11, t = 2 mod 5}
09:47 < andytoshi> run this through the regular CRT which is super fast (ext. euclid's algo on 11 and 5) and out pops t = 68. and sure enough, 68 mod 11 = 2, which is the answer
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09:52 < tromp> i used CRT to compute the number of legal Go positions...
09:52 < Eliel_> tromp: now I'm curious if that's where the theorem got it's initial inspiration :)
09:53 < tromp> CRT is way older than go counting:(
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10:43 < andytoshi> the eqn for b1 on the bottom of page 3 is wrong. if you use it you get a different value in their example (and you fail to solve the discrete log)
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11:22 < andytoshi> honestly i think the proof of lemma 2 is wrong, they handwaved a step and mixed up n*phi(q) - 1 and (n - 1)*phi(q) .... then when they were doing their "numerical example" they just fudged the numbers to make it work
11:23 < andytoshi> i.e. the paper is wrong
11:24 < andytoshi> maaku: ^
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11:53 < maaku> Thanks
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--- Log closed Mon Aug 29 00:00:39 2016