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11:09 < jeremyrubin> Does anyone have any thoughts about introducing an opcode for verifying either the result of a multiplication (or ideally!) a division?
11:10 < jeremyrubin> If you have an opcode which permits checking if n (32 bit) divides m (64 bit), then a nice thing you can do is express circuits by "prime painting" conditions.
11:10 < jeremyrubin> e.g., if I have logical conditions A B C D
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11:11 < jeremyrubin> I can assign A = 2, B=3, C=5, D=7
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11:11 < jeremyrubin> I can then have an IF branch whcih is like this
11:12 < jeremyrubin> <n> | DUP2 3 DIVIDES IF 7 DIVIDES {condition B} ENDIF
11:13 < jeremyrubin> This expresses tightly that we can provide a witness <n> which is the product of all the conditions we wish to execute
11:14 < jeremyrubin> And that each condition can express the other conditions it relies on
11:14 < sipa> can't you do the same with bitmasks?
11:15 < jeremyrubin> I think it depends on what the bitmask operations look like
11:16 < jeremyrubin> I don't think it can be done as compactly, but I might be wrong there
11:16 < jeremyrubin> I mean logically I think you are fitting the same amount of info in
11:16 < jeremyrubin> But with primes
11:17 < jeremyrubin> the product of the first 168 primes is less than 64 bits
11:17 < jeremyrubin> so it seems to be more efficient
11:17 < jeremyrubin> If [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467,
11:17 < jeremyrubin> 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997] is correct
11:18 < sipa> that seems impossible, unless all the first 168 primes are less than 2
11:18 < jeremyrubin> hmm
11:18 < jeremyrubin> yeah I agree
11:18 < jeremyrubin> hang on
11:18 < jeremyrubin> oh
11:19 < jeremyrubin> silent overflows in python code
11:19 < sipa> log(product(list(primes(1,1000)))+0.0)/log(2.0)
11:19 < sipa> 1379.57030186226
11:20 < jeremyrubin> Ok this makes more sense now
11:21 < jeremyrubin> You can get [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
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11:21 < sipa> i still don't understand why you need something multiplicative
11:22 < sipa> maybe rather, i don't understand what you're trying to do, but i suspect it can either be done with bitmasking, or using weighted thresholds
11:23 < jeremyrubin> Oh I agree it can be done with masking
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11:26 < jeremyrubin> Just interesting if theres a more general primitive (division) that gets a similar outcome.
11:28 < jeremyrubin> I'm also curious theoretically what the space differences are across use cases
11:28 < jeremyrubin> I guess that depends heavily on encoding schemes for bit masks
11:29 < jeremyrubin> There's also some other weird math things with the division stuff
11:29 < sipa> you can see this construction as an encoding of a list of integers, e.g. [4 6 0 3 0 0 0 ...] would be turned into 2^4 * 3^6 * 5^0 * 7^3 * ...
11:29 < jeremyrubin> Like if you know you're k-bounded on condition anding
11:29 < sipa> where you can efficiently prove that one list is in every position larger or equal than another list's pointwise positions
11:30 < jeremyrubin> Right, interesting. So bitmasking is really the same thing even as a mathematical object
11:30 < sipa> if you don't need these integers to be potentiallly arbitrarily large, you can use something simpler
11:30 < sipa> bitmasking is the same, but with numbers restricted to 0..1
11:30 < jeremyrubin> Correct
11:30 < sipa> and a much more compact encoding
11:31 < jeremyrubin> It's the same I beleive
11:31 < jeremyrubin> Well I guess not
11:31 < sipa> of course not
11:31 < sipa> with bitmasking you can encode 64 variables in 64 bits
11:31 < jeremyrubin> Because of the multiple primes in a factor taking up space
11:31 < sipa> exactly
11:32 < jeremyrubin> So I guess the other side of it that's interesting too is that if your logic is bounded
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11:32 < jeremyrubin> e.g., at most k-clauses
11:32 < sipa> i believe you can use this instead
11:32 < sipa> say you want up to 7 clauses
11:33 < sipa> which fits in 3 bits
11:33 < sipa> then you use a bitmask, but encode every variable in 3 bits
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11:33 < sipa> use normal addition/subtraction for testing
11:33 < sipa> and bit fiddling tricks to verify that none of the individual groups overflowed
11:33 < sipa> you may need 4 bits per variable
11:33 < jeremyrubin> So I guess I'm curious what the encoding is of the masks for querying.
11:34 < jeremyrubin> if it ends up being more efficient or not
11:34 < jeremyrubin> One of the other interesting properties I think is if you bring a curve into the picture
11:34 < jeremyrubin> because then you can probably embed arbitrary primes into it and show that you know two points
11:35 < sipa> i'm skeptical about that
11:35 < sipa> modulo the order of the curve, there are no primes (every element is invertible)
11:36 < jeremyrubin> hmm
11:37 < sipa> if you could, it would sound like you'
11:37 < jeremyrubin> I sort of disagree
11:37 < sipa> if you could, it would sound like you'd have a compact EC-based accumulator
11:37 < jeremyrubin> Because, for example
11:37 < jeremyrubin> yes
11:37 < jeremyrubin> In this case it's permissable to have a trusted setup BTW
11:37 < sipa> i believe that is known to not be possible, or at least lots of people have tried :)
11:38 < jeremyrubin> Because you would be proving a script compilation for a group of people let's say
11:38 < sipa> yes, even with trusted setup
11:38 < jeremyrubin> I thought those were down to like ~300 bits?
11:38 < sipa> you mean the hyperelliptic curve stuff?
11:39 < sipa> if you call that EC, sure, but it relies on a curve of unknown order
11:40 < sipa> https://eprint.iacr.org/2020/196
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23:35 < jeremyrubin> So interestingly you can actually relatively efficiently for the small numbers unroll division into conditional subtraction in script TODAY
23:35 < jeremyrubin> But I think for accomplishing something close to the thing I'm interested in, there's some better scripts yet
23:36 < jeremyrubin> Coming up with some interesting gadgets, for such scripts
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--- Log closed Wed Apr 22 00:00:47 2020