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From: Gregory Maxwell <greg@xiph.org>
Date: Mon, 9 Jul 2018 16:21:59 +0000
Message-ID: <CAAS2fgTmEaM8XStQ3xQns2JpS3upJ==MQv65YfxYFtWKFNXi5Q@mail.gmail.com>
To: Erik Aronesty <erik@q32.com>, 
	Bitcoin Protocol Discussion <bitcoin-dev@lists.linuxfoundation.org>
Content-Type: text/plain; charset="UTF-8"
Subject: Re: [bitcoin-dev] Multiparty signatures
Precedence: list

On Mon, Jul 9, 2018 at 3:02 PM, Erik Aronesty via bitcoin-dev
<bitcoin-dev@lists.linuxfoundation.org> wrote:
> and where H(g*x) can
> be considered their public index for the purposes of Shamir polynomial
> interpolation

This is isomorphic to the insecure musig variant where keys are
blinded by H(g*x) instead of a commitment to all keys. It is insecure
because it vulnerable to an attacker knowing a victim pubkey P  who
uses wagner's algorithim to solve a random modular subset sum problem:
-1H(P) = H(aP)/a + H(bP)/b + H(cP)/c + ... for some a,b,c...  then
claiming to be participants with keys aP, bP, cP, ..., xG (their own
key) and canceling out key P, allowing the value to just be signed for
with their key alone.

AFAICT your suggestion is using simple multiplication in the place of
a cryptographic hash.  E.g.  you have just suggested a schnorr
signature where H() is  just r*m in the field of size n. It doesn't
have any new properties about how you can use it. The same linearities
do and don't apply as the normal schnorr construction, but for any of
the security proofs to hold we'd have to believe that multiplication
in the field of n is a suitable random oracle-- which is not very
plausible.