--- Log opened Mon Mar 23 00:00:30 2020
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08:05 < kanzure> "Verifiable secret redistribution for threshold sharing schemes" http://www.cs.cmu.edu/~wing/publications/Wong-Wing02b.pdf
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08:54 < bsm117532> ^^^ That paper has a hole: their Pedersen commitments omit the "hiding" element.
08:54 < bsm117532> But I still think it's a nice demonstration of how redistributable, verifiable secret sharing can work.
08:55 < bsm117532> The consequence of omitting hiding is that if you know (or can guess) a few bits of the secret, you can derive some bits of the polynomial coefficients.
08:57 < bsm117532> See CHURP instead if you're interested in this idea: https://eprint.iacr.org/2019/017, or encrypt your secret so that it is uniformly random (adding a hiding element).
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09:04 < bsm117532> Is Z_{4p+1} an appropriate ring for committing to values in Z_p?  (bitcoin private keys, where p = bitcoin's prime).
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09:16 < sipa> bsm117532: what does that mean?
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09:20 < bsm117532> The above paper requires Pedersen commitments to the secret values. I'm wondering if the prime 4p+1 is generally appropriate for commitments to bitcoin private keys. (independent of that paper -- might be used in other ways)
09:21 < sipa> is that a multiplication-mod-prime group? i haven't read the paper
09:21 < sipa> if so, 4*p+1 woukd be way too small
09:21 < bsm117532> https://tlu.tarilabs.com/cryptography/bulletproofs-protocols/MainReport.html#pedersen-commitments-and-elliptic-curve-pedersen-commitments
09:22 < bsm117532> Such things require a prime 2*p+1, but for the bitcoin prime, 2*p+1 isn't prime, but 4*p+1 is.
09:22 < sipa> oh that's a simplification
09:22 < sipa> in reality you will always use EC groups
09:22 < sipa> instead of DSA-style group
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09:23 < bsm117532> Sure. The only reason I ask is the above paper uses exponentiated Pedersen commitments, not EC commitments.
09:24 < bsm117532> I was fooling with this years ago, have been wondering about 4p+1 ever since, but never satisfied myself that 4p+1 was secure. Yes it's small. r=(4p+1)+1 has a 133-bit factor. I'm not sure how to use that though...
09:27 < sipa> you need a 3000-bit group for 12i-bit sexurity
09:27 < sipa> you need a 3000-bit group for 128-bit security
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09:30 < bsm117532> But this is just a Pedersen commitment, not RSA. how would you use a too-small prime to break a Pedersen commitment?
09:31 < sipa> DL happens to be exactly as hard as factorization
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09:39 < bsm117532> Ok, but how ;-)
09:40 < sipa> if DL is broken, the commitment is broken, no?
09:40 < sipa> otherwise, you could just a much simpler group if DL hardness was not required
09:45 < bsm117532> I accept your argument, but I'm looking for a procedure to extract the committed value, just to help me understand. ;-)
09:46 < sipa> pedersen commitments are information theoretically hiding (if the blinding factor is random), there is no way to.extract
09:46 < sipa> but if DL is brokem you can open them to whatever value you like
09:47 < sipa> if that's ok to you, you could use a+r as a commitment to a :p
09:47 < sipa> or the constant 11
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10:27  * bsm117532 scurries off to figure out how to break discrete log commitments...
10:28 < sipa> say you commit to a using an EC commitment with generators G and H, in a group in which DL is easy
10:29 < sipa> your commitment is aG+rH
10:29 < sipa> you'd like to open it to value c instead
10:29 < sipa> so you're trying to find s such that cG+sH = aG+bH
10:30 < sipa> s = (aG+bH-cG)/H
10:30 < sipa> where / is your DL algorithm
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11:59 < bsm117532> sipa: (segue from Twitter) The bad idea in my head is to do a multisignature with (e.g. 2-of-3) P = P_1 P_2 + P_2 P_3 + P_1 P_3. This is a monotone boolean function on the pubkeys with + as an OR operation and * as an AND operation.
11:59 < bsm117532> (Insert hashes in front of the terms as necessary to commit to this combination being allowed)
11:59 < bsm117532> Parties 1 and 2 could then construct a signature by using the ideas from 2p-ECDSA (multiplicative aggregate keys), and possibly abusing the Schnorr related  key attack to remove the extra terms.
11:59 < bsm117532> Is something along these lines impossible? Or is it just that no one has described how to do it?
12:00 < sipa> well that depends on what your OR operation is
12:00 < sipa> if it's a Merkle tree, that's certainly been described before
12:01 < sipa> if it's SSS, that's verifiable secret sharing based threshold crypto
12:04 < sipa> well, not really; the SSS is doing both the ANDs and ORs
12:05 < sipa> but there are schemes (somewhat related to VSS) that can translate any monotone boolean function over keys to a setup for that policy
12:08 < bsm117532> I'm familiar with all the SSS stuff, I'm trying to find a non-SSS way to do it. ;-)
12:09 < bsm117532> OR operation is using (abusing) related key attacks in some way.
12:09 < sipa> make a Merkle tree where every leaf is a MuSig of some subset of the keys; and you have one leaf for each of the (n choose k) combinations
12:09 < sipa> done
12:09 < sipa> i don't understand the relation with related key attacks
12:10 < bsm117532> Yep, that Merkle construction certainly works...
12:11 < sipa> this of course has signatures that scale with O(log(n choose k))
12:11 < bsm117532> related key: just saying it's in general possible to sign for P+a if you have the private keys for P. (you called it a "related key attack" in bip-schnorr) Or pubkey tweak, or whatever.
12:12 < sipa> bsm117532: that's VSS; the signers effectively use shamir shares to reconstruct the (partial signature for) the secret keys they are missing; the linearity of the keys and signatures means that's enough
12:13 < bsm117532> Sure you can do that with VSS, I know.
12:13 < sipa> then i don't know what you're asking for
12:15 < bsm117532> I'm asking for an algorithm for 2-of-3 parties to construct a signature for the aggregate pubkey P = P_1 P_2 + P_2 P_3 + P_1 P_3, without using Shamir sharing.
12:15 < bsm117532> (and I don't think one exists, probably because no one has tried?)
12:16 < bsm117532> Or, some reason this is a terrible idea...
12:16 < sipa> what does the multiplication even mean?
12:16 < sipa> you can't multiply keys
12:16 < sipa> i know you mean "something representing and", but if you're not concrete then the question is meaningless
12:17 < bsm117532> I mean P = p_1 p_2 G + p_2 p_3 G + p_1 p_3 G for lower case p being private keys
12:17 < sipa> ah!
12:17 < bsm117532> Sorry, I was sloppy
12:17 < sipa> so your multiplication is effectively DH
12:17 < bsm117532> Yes.
12:17 < sipa> now you're going to need proofs of DLEQ
12:17 < sipa> to compute the aggregate
12:18 < sipa> how would what you're trying to build be better than VSS?
12:19 < bsm117532> It probably wouldn't because of the factorial number of terms involved. I'm just curious...since this is what was in my head naively as "native threshold Schnorr".
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12:25 < bsm117532> I also know some optimizations for polynomial representation that I might dig in to to reduce that factorial complexity, if I thought the idea in general was workable...
12:26 < yanmaani> \
12:26 < yanmaani> oops
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--- Log closed Tue Mar 24 00:00:31 2020