Actually, it looks like in order to compute a multiparty signature you will need to broadcast shares of r first, so it's not offline :(
It is still seems, to me, to be a simpler mechanism than musig - with security assumptions that match the original Schnorr construction more closely, and should therefore be easier to prove secure in a multiparty context. Shamir/Schnorr threshold multi-signature scheme: Each party: - Has a public key g*x', where x' is their private key, and where H(g*x) can be considered their public index for the purposes of Shamir polynomial interpolation - Rolls a random k' and compute r' = g*k' - Broadcast r' as a share - Computes g*k, via lagrange interpolation across shares. At this point k is not known to any party unless Shamir is vulnerable or DL is not hard - Computes e' = H(M) * r' - Computes s' = k'-x*e' - Share of signature is (s', e') Verification is the same as Scnhorr, but only after using interpolation to get the needed (s, e, g*x) from shares of s', e' and g*x': - Using lagrange interpolation, compute the public key g*x - Again, using lagrange interpolation, compute (s, e) - Verify the signature as per standard Schnorr Security assumptions: - Because this is not additive, and instead we are using Shamir combination, the additional blinding and masking steps of musig are not needed to create a secure scheme. - The scheme is the same as Schnorr otherwise - The only thing to prove is that H(M) * r does not reveal any information about k ... which relies on the same DL assumptions as Bitcoin itself - Overall, this seems, to me at least, to have a smaller attack surface because there's fewer moving parts On Mon, Jul 9, 2018 at 8:24 AM, Erik Aronesty <e...@q32.com> wrote: > I was hoping that nobody in this group saw an obvious problem with it then > I'd sit down and try to write up a paper. > > Not that hard to just reuse the work done on schnorr. And demonstrate > that there are no additional assumptions. > > On Mon, Jul 9, 2018, 12:40 AM Pieter Wuille <pieter.wui...@gmail.com> > wrote: > >> On Sun, Jul 8, 2018, 21:29 Erik Aronesty <e...@q32.com> wrote: >> >>> Because it's non-interactive, this construction can produce multisig >>> signatures offline. Each device produces a signature using it's own >>> k-share and x-share. It's only necessary to interpolate M of n shares. >>> >>> There are no round trips. >>> >>> The security is Shamir + discrete log. >>> >>> it's just something I've been tinkering with and I can't see an obvious >>> problem. >>> >>> It's basically the same as schnorr, but you use a threshold hash to fix >>> the need to be online. >>> >>> Just seems more useful to me. >>> >> >> That sounds very useful if true, but I don't think we should include >> novel cryptography in Bitcoin based on your not seeing an obvious problem >> with it. >> >> I'm looking forward to seeing a more complete writeup though. >> >> Cheers, >> >> -- >> Pieter >> >> >>
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