Re: [bitcoin-dev] Multiparty signatures

[Erik Aronesty <erik@q32.com>]

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