[Cfrg] Fwd: 25519 naming

David Leon Gil <coruus@gmail.com> Wed, 27 August 2014 18:15 UTC

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From: David Leon Gil <coruus@gmail.com>
Date: Wed, 27 Aug 2014 14:15:33 -0400
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Subject: [Cfrg] Fwd: 25519 naming
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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 of
> 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 coördinates 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 = u^3 + 486662 u^2 + u over
[Fq25519].
- [...] Curve25519 is birationally equivalent to an Edwards curve,
specifically x^2 + y^2 = 1 + (121665/121666) x^2 y^2; the equivalence is x
= √486664 u / v and y = (u − 1)/(u + 1).
- [T]his Edwards curve is isomorphic to −x^2 + y^2 = 1 − (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