--- Log opened Tue Dec 07 00:00:54 2021
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05:42 < roconnor> andytoshi: So none of the rings of characters on the recover shares disc are rotations of another at a different radius.
05:43 < roconnor> And I'm 75% sure that permuting the outer labels won't help because you  cannot take two unaligned rings and permute them in identical ways to make them aligned.
05:44 < roconnor> This is unlike the addition discs where the rings all come in pairs which are rotations of each other.
05:49 < andytoshi> roconnor: hmm, yeah, i buy that argument
05:50 < roconnor> exchanging the order of arguments to the recover shares disc doesn't help either.
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05:54 < roconnor> mabye I'm wrong though, because my argument seems to say the multiply disc cannot be optimized either ....
05:55 < andytoshi> i would've guessed that every row on the multiply disc is a rotation of another?
05:55 < roconnor> me too
05:58 < roconnor> oh I see, my argument isn't correct.
05:59 < roconnor> you don't permute the two rings in the same way.  The rings are offset by some amount so one ring gets permuted on one way, and nother ring gets permuted in a different way, probably conjugated by a rotation.
06:00 < roconnor> So, in the case of multiplication, if you permute the outer ring into a proper order, then all the rings become rotations of each other, but it doesn't hold for general orders.
06:01 < andytoshi> ah! ofc
06:01 < roconnor> and 32! is too large for me to blindly try every permuation for the recover shares disc.  .... I would have to use math.
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06:03 < andytoshi> i think you could plausibly dynamic program it, by getting the first two characters beside each other, then the third, etc., checking that just those characters' results wind up being rotated vs each other
06:14 < roconnor> Math suggests that if the secret were at Q, then using powers of alpha on the outer ring would align all the recover shares rings.
06:19 < roconnor> If the secret is at S, then you put the outer ring in powers of alpha `xor` S.
06:19 < roconnor> and all the rings are rotations of each other.
06:19 < roconnor> tada
06:20 < sipa> i see "powers of alpha" en cyclic groups, and i now wonder if you've somehow recreated something BCH-like on paper
06:23 < roconnor> This suggest everything can be done with circular slide rulers, except for additions, which can be done with a smaller volvelle of half the area.
06:31 < andytoshi> roconnor: cool!!
06:31 < andytoshi> sipa: i think this is just the observation that all fields' multiplicative groups are cyclic
06:32 < andytoshi> BCH codes involve modding out by a non-irreducible polynomial so the resulting structure isn't a field, so the reasons for its cyclic structure are more "accidental"
07:58 < roconnor> And presumably we don't have to use power of alpha, but rather any generator will do, which is everthing that isn't 0 or 1.
08:00 < sipa> you're lucky that 31 is prime
08:01 < sipa> if it wasn't, we'd need to throw out pretty much all math, i fear
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08:31 < roconnor> I don't know about that.  I mean we wouln't have a choice of 30 generators, but I kinda think things would still work.
08:32 < roconnor> andytoshi: It's still morning, but I'm starting to get the feeling that we don't have to put the cosets thingys 180 degrees apart.
08:32 < roconnor> and that we can use basically any order, including alphabetical.
08:33 < roconnor> And we just have to place holes in the top disc appropriately.
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08:42 < roconnor> at each ring radius there will be two punches and some angle to each other.
08:42 < roconnor> *at some angle
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08:59 < roconnor> Okay, I don't think the non-180 thing works.
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09:13 < andytoshi> yeah, i think it has to be 180 ... though i am musing about the order-5 frobenius automorphism and 72-degree turns
09:14 < andytoshi> regarding "any generator will work for the multiplication wheels" yep, so we may want to look through all 30 of these to see what's most "natural"
09:16  * gene loves maths jargon sounding like rick and morty aliens
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09:23 < roconnor> andytoshi: unfortunately 72 degree turns isn't a thing.
09:24 < andytoshi> roconnor: well, Q and P would need to be treated separately, then the remaining 30 elements could be split into 6 groups of 5 (or 5 groups of 6?)
09:25 < andytoshi> oh, i see, the outer set of digits can't be separated like that
09:25 < andytoshi> unless we dropped them from the set of things you can add by :P which is probably ok for Q but would be really weird for P
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09:46 < andytoshi> yeah ok i don't think frobenius gets us anything ... it lets us turn a+b=c into a^2+b^2=c^2 but we actually don't want to vary a and b simultaneously
09:50 < roconnor> I couldn't figure out anything either.
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