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From: Erik Aronesty <erik@q32.com>
Date: Thu, 13 Sep 2018 16:20:36 -0400
Message-ID: <CAJowKg+0uOZ5_ryFit6-GW_fEbkXwBU8m7VAAOxgZAzP_5rF8A@mail.gmail.com>
To: apoelstra@wpsoftware.net
Content-Type: multipart/alternative; boundary="0000000000007515910575c67147"
Cc: Bitcoin Protocol Discussion <bitcoin-dev@lists.linuxfoundation.org>
Subject: Re: [bitcoin-dev] Schnorr signatures BIP
Precedence: list

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The paper refers to either:

  a) building up threshold signatures via concatenation, or. implicitly -
in Bitcoin -
  b) by indicating that of M of N are valid, and requiring a validator to
validate one of the permutations of M that signed - as opposed to a scheme,
like a polynomial function, where the threshold is built in to the system.

Maybe there's another mechanism in there that I'm not aware of - because
it's just too simple to mention?

- Erik


On Thu, Sep 13, 2018 at 2:46 PM Andrew Poelstra <apoelstra@wpsoftware.net>
wrote:

> On Tue, Sep 11, 2018 at 01:37:59PM -0400, Erik Aronesty via bitcoin-dev
> wrote:
> > - Musig, by being M of M, is inherently prone to loss.
> >
>
> It has always been possible to create M-of-N threshold MuSig signatures
> for any
> M, N with 0 < M =E2=89=A4 N. This is (a) obvious, (b) in our paper, (c)
> implemented at
>
>
> https://github.com/apoelstra/secp256k1/blob/2018-04-taproot/src/modules/m=
usig/main_impl.h
>
> --
> Andrew Poelstra
> Research Director, Mathematics Department, Blockstream
> Email: apoelstra at wpsoftware.net
> Web:   https://www.wpsoftware.net/andrew
>
> "Make it stop, my love; we were wrong to try
>  Never saw what we could unravel in traveling light
>  Nor how the trip debrides like a stack of slides
>  All we saw was that time is taller than space is wide"
>        --Joanna Newsom
>
>

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<div dir=3D"ltr"><div>The paper refers to either:</div><div><br></div><div>=
=C2=A0 a) building up threshold signatures via concatenation, or. implicitl=
y - in Bitcoin - <br></div><div>=C2=A0 b) by indicating that of M of N are =
valid, and requiring a validator to validate one of the permutations of M t=
hat signed - as opposed to a scheme, like a polynomial function, where the =
threshold is built in to the system.</div><div><br></div><div>Maybe there&#=
39;s another mechanism in there that I&#39;m not aware of - because it&#39;=
s just too simple to mention?</div><div><br></div><div>- Erik<br></div><div=
><br></div><div><br></div><div><br></div><div><br></div><div><br></div></di=
v><br><div class=3D"gmail_quote"><div dir=3D"ltr">On Thu, Sep 13, 2018 at 2=
:46 PM Andrew Poelstra &lt;<a href=3D"mailto:apoelstra@wpsoftware.net" targ=
et=3D"_blank">apoelstra@wpsoftware.net</a>&gt; wrote:<br></div><blockquote =
class=3D"gmail_quote" style=3D"margin:0 0 0 .8ex;border-left:1px #ccc solid=
;padding-left:1ex">On Tue, Sep 11, 2018 at 01:37:59PM -0400, Erik Aronesty =
via bitcoin-dev wrote:<br>
&gt; - Musig, by being M of M, is inherently prone to loss.<br>
&gt;<br>
<br>
It has always been possible to create M-of-N threshold MuSig signatures for=
 any<br>
M, N with 0 &lt; M =E2=89=A4 N. This is (a) obvious, (b) in our paper, (c) =
implemented at<br>
<br>
<a href=3D"https://github.com/apoelstra/secp256k1/blob/2018-04-taproot/src/=
modules/musig/main_impl.h" rel=3D"noreferrer" target=3D"_blank">https://git=
hub.com/apoelstra/secp256k1/blob/2018-04-taproot/src/modules/musig/main_imp=
l.h</a> <br>
<br>
-- <br>
Andrew Poelstra<br>
Research Director, Mathematics Department, Blockstream<br>
Email: apoelstra at <a href=3D"http://wpsoftware.net" rel=3D"noreferrer" ta=
rget=3D"_blank">wpsoftware.net</a><br>
Web:=C2=A0 =C2=A0<a href=3D"https://www.wpsoftware.net/andrew" rel=3D"noref=
errer" target=3D"_blank">https://www.wpsoftware.net/andrew</a><br>
<br>
&quot;Make it stop, my love; we were wrong to try<br>
=C2=A0Never saw what we could unravel in traveling light<br>
=C2=A0Nor how the trip debrides like a stack of slides<br>
=C2=A0All we saw was that time is taller than space is wide&quot;<br>
=C2=A0 =C2=A0 =C2=A0 =C2=A0--Joanna Newsom<br>
<br>
</blockquote></div>

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