sipa changed the topic of #bitcoin-wizards to: This channel is for discussing theoretical ideas with regard to cryptocurrencies, not about short-term Bitcoin development | http://bitcoin.ninja/ | This channel is logged. | For logs and more information, visit http://bitcoin.ninja
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<jeremyrubin> Does anyone have any thoughts about introducing an opcode for verifying either the result of a multiplication (or ideally!) a division?
<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.
<jeremyrubin> e.g., if I have logical conditions A B C D
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<jeremyrubin> I can assign A = 2, B=3, C=5, D=7
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<jeremyrubin> I can then have an IF branch whcih is like this
<jeremyrubin> <n> | DUP2 3 DIVIDES IF 7 DIVIDES {condition B} ENDIF
<jeremyrubin> This expresses tightly that we can provide a witness <n> which is the product of all the conditions we wish to execute
<jeremyrubin> And that each condition can express the other conditions it relies on
<sipa> can't you do the same with bitmasks?
<jeremyrubin> I think it depends on what the bitmask operations look like
<jeremyrubin> I don't think it can be done as compactly, but I might be wrong there
<jeremyrubin> I mean logically I think you are fitting the same amount of info in
<jeremyrubin> But with primes
<jeremyrubin> the product of the first 168 primes is less than 64 bits
<jeremyrubin> so it seems to be more efficient
<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,
<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
<sipa> that seems impossible, unless all the first 168 primes are less than 2
<jeremyrubin> hmm
<jeremyrubin> yeah I agree
<jeremyrubin> hang on
<jeremyrubin> oh
<jeremyrubin> silent overflows in python code
<sipa> log(product(list(primes(1,1000)))+0.0)/log(2.0)
<sipa> 1379.57030186226
<jeremyrubin> Ok this makes more sense now
<jeremyrubin> You can get [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
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<sipa> i still don't understand why you need something multiplicative
<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
<jeremyrubin> Oh I agree it can be done with masking
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<jeremyrubin> Just interesting if theres a more general primitive (division) that gets a similar outcome.
<jeremyrubin> I'm also curious theoretically what the space differences are across use cases
<jeremyrubin> I guess that depends heavily on encoding schemes for bit masks
<jeremyrubin> There's also some other weird math things with the division stuff
<jeremyrubin> Like if you know you're k-bounded on condition anding
<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 * ...
<sipa> where you can efficiently prove that one list is in every position larger or equal than another list's pointwise positions
<jeremyrubin> Right, interesting. So bitmasking is really the same thing even as a mathematical object
<sipa> if you don't need these integers to be potentiallly arbitrarily large, you can use something simpler
<sipa> bitmasking is the same, but with numbers restricted to 0..1
<jeremyrubin> Correct
<sipa> and a much more compact encoding
<jeremyrubin> It's the same I beleive
<jeremyrubin> Well I guess not
<sipa> of course not
<sipa> with bitmasking you can encode 64 variables in 64 bits
<jeremyrubin> Because of the multiple primes in a factor taking up space
<sipa> exactly
<jeremyrubin> So I guess the other side of it that's interesting too is that if your logic is bounded
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<jeremyrubin> e.g., at most k-clauses
<sipa> i believe you can use this instead
<sipa> say you want up to 7 clauses
<sipa> which fits in 3 bits
<sipa> then you use a bitmask, but encode every variable in 3 bits
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<sipa> use normal addition/subtraction for testing
<sipa> and bit fiddling tricks to verify that none of the individual groups overflowed
<sipa> you may need 4 bits per variable
<jeremyrubin> So I guess I'm curious what the encoding is of the masks for querying.
<jeremyrubin> if it ends up being more efficient or not
<jeremyrubin> One of the other interesting properties I think is if you bring a curve into the picture
<jeremyrubin> because then you can probably embed arbitrary primes into it and show that you know two points
<sipa> i'm skeptical about that
<sipa> modulo the order of the curve, there are no primes (every element is invertible)
<jeremyrubin> hmm
<sipa> if you could, it would sound like you'
<jeremyrubin> I sort of disagree
<sipa> if you could, it would sound like you'd have a compact EC-based accumulator
<jeremyrubin> Because, for example
<jeremyrubin> yes
<jeremyrubin> In this case it's permissable to have a trusted setup BTW
<sipa> i believe that is known to not be possible, or at least lots of people have tried :)
<jeremyrubin> Because you would be proving a script compilation for a group of people let's say
<sipa> yes, even with trusted setup
<jeremyrubin> I thought those were down to like ~300 bits?
<sipa> you mean the hyperelliptic curve stuff?
<sipa> if you call that EC, sure, but it relies on a curve of unknown order
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