[bitcoin-dev] Multiparty signatures

[Dan Robinson]

Can you please clarify which terms in that description are elliptic curve
points, and which are scalars?
On Mon, Jul 9, 2018 at 11:10 AM Erik Aronesty via bitcoin-dev <
bitcoin-dev at lists.linuxfoundation.org> wrote:

> 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 <erik at 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.wuille at gmail.com>
>> wrote:
>>
>>> On Sun, Jul 8, 2018, 21:29 Erik Aronesty <erik at 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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