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From: David Leon Gil <coruus@gmail.com>
Date: Wed, 27 Aug 2014 14:15:33 -0400
Message-ID: <CAA7UWsUS1vGXwuQtfvgk3qVYpzEQVqQCFYEkoyRO4m4Sk2sN-A@mail.gmail.com>
To: "cfrg@irtf.org" <cfrg@irtf.org>
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Archived-At: http://mailarchive.ietf.org/arch/msg/cfrg/BSLb_4Mc3ht32x4NkMwjn6JTCGc
Subject: [Cfrg] Fwd:  25519 naming
Precedence: list

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(Yes; I significantly misspoke in my first paragraph: indeed,
different geometries. The perils of curves before coffee.)

On Wednesday, August 27, 2014, Watson Ladd <watsonbladd@gmail.com> wrote:
>
>  To avoid all this verbiage, which everyone understands (with the usual
> criteria about what "everyone" means) we speak in the informal language o=
f
> coordinates, and then mentally translate into the language of EGA. I don'=
t
> see what's missing from DJBs email in this regard.
>
'Everyone', for our purposes, should be the 'everyone' who reads RFCs.

These two things are quite different:

- Changing co=C3=B6rdinates on the *same* curve (e.g., Jacobian to affine).
- Transferring points to another equivalent curve (e.g., Montgomery to
Weierstrass).

My posit is that eliding this difference is (a small part of) the *cause*
of the confusion djb notes.

It is easy enough to be explicit about the latter. E.g., from the Ed25519
paper (line breaks inserted):

- Curve25519 ... is the Montgomery curve v^2 =3D u^3 + 486662 u^2 + u over
[Fq25519].
- [...] Curve25519 is birationally equivalent to an Edwards curve,
specifically x^2 + y^2 =3D 1 + (121665/121666) x^2 y^2; the equivalence is =
x
=3D =E2=88=9A486664 u / v and y =3D (u =E2=88=92 1)/(u + 1).
- [T]his Edwards curve is isomorphic to =E2=88=92x^2 + y^2 =3D 1 =E2=88=92 =
(121665/121666)
x^2 y^2 . . . .

I'd suggest that talking in this way is better than talking about different
curves as 'coordinate systems'.

-dlg

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<div dir=3D"ltr"><div class=3D"gmail_quote">(Yes; I significantly misspoke =
in my first paragraph: indeed, different=C2=A0geometries.=C2=A0The perils o=
f curves=C2=A0before coffee.)<div class=3D""><br>On Wednesday, August 27, 2=
014, Watson Ladd &lt;<a href=3D"mailto:watsonbladd@gmail.com" target=3D"_bl=
ank">watsonbladd@gmail.com</a>&gt; wrote:=C2=A0<blockquote class=3D"gmail_q=
uote" style=3D"margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1e=
x">


<div dir=3D"ltr">



<p>To avoid all this verbiage, which everyone understands (with the usual c=
riteria about what &quot;everyone&quot; means) we speak in the informal lan=
guage of coordinates, and then mentally translate into the language of EGA.=
 I don&#39;t see what&#39;s missing from DJBs email in this regard.</p>


</div></blockquote></div>&#39;Everyone&#39;, for our purposes, should be th=
e &#39;everyone&#39;=C2=A0who reads RFCs.<br><div><br></div><div>These=C2=
=A0two things are quite different:</div><div><br></div><div>- Changing=C2=
=A0co=C3=B6rdinates on the *same* curve (e.g., Jacobian to=C2=A0affine).</d=
iv>


<div>- Transferring=C2=A0points=C2=A0to another equivalent curve (e.g., Mon=
tgomery to Weierstrass).</div><div><br></div><div>My posit is that eliding=
=C2=A0this difference is (a small=C2=A0part of)=C2=A0the *cause* of the con=
fusion djb notes.</div>


<div><br></div><div>It is easy=C2=A0enough to be explicit=C2=A0about the la=
tter.=C2=A0E.g., from the Ed25519 paper (line breaks inserted):</div><div><=
br></div><div>-=C2=A0Curve25519 ...=C2=A0is the Montgomery curve v^2 =3D u^=
3 + 486662 u^2 + u over [Fq25519].</div>


<div>-=C2=A0[...] Curve25519 is birationally equivalent to an Edwards curve=
, specifically x^2 + y^2 =3D 1 + (121665/121666) x^2 y^2; the equivalence i=
s x =3D =E2=88=9A486664 u / v and y =3D (u =E2=88=92 1)/(u + 1).</div><div>=
-=C2=A0[T]his Edwards curve is isomorphic to =E2=88=92x^2 + y^2 =3D 1 =E2=
=88=92 (121665/121666) x^2 y^2 . . . .</div>


<div><br></div><div>I&#39;d suggest that talking in this way is better than=
 talking about different curves as=C2=A0&#39;coordinate systems&#39;.</div>=
<span class=3D"HOEnZb"><font color=3D"#888888"><div><br></div><div>-dlg</di=
v>
</font></span></div><br></div>

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