Re: [Cfrg] Edwards ladder

[Robert Ransom <[e-mail]>]

On 12/2/14, Watson Ladd <[e-mail]> wrote:
> Dear all,
>
> The formulas on the EFD for a y-coordinate only Edwards ladder require
> d to be a square. They are slightly more efficient than the Montgomery
> ladder when squaring is specially optimized. Unfortunately, the
> Edwards curve formulas we are considering don't have d square.

In order for the ladder formula shown on
<http://hyperelliptic.org/EFD/g1p/auto-edwards-yzsquared.html> to be
even close to more efficient, s must be chosen to be a small integer;
r is then a ratio of small integers and 1*r requires two small-integer
multiplies.

Mike Hamburg tried that formula with an earlier version of his Ed448
software, and found that they were slower than the Montgomery-form
ladder.


In addition, that formula has an exceptional case when Y1=0 (the
affine points of order 4), which is relatively harmless for
cryptographic purposes but makes implementation-distinguishing attacks
more likely and makes a mathematically correct implementation more
complex than a merely interoperable implementation.

I don't see the point of investigating those formulas further.


(If anyone else wants to try them, I found a twist-secure curve
optimized for that formula over the same field as Curve3617; the curve
parameter's value was 2170.  I think the parameter was s.)


> The relatively little amount of thought I've given to this matter
> hasn't produced an isogeny that makes d square. Furthermore, picking d
> square makes the addition laws not complete: it's not possible to have
> both the efficient ladder and a complete addition law simultaneously
> when using Edwards coordinates.

You can map a complete Edwards curve with cofactor 8 (or divisible by
8) to an Edwards curve with square d by using the E_d -> L_d map shown
in <https://eprint.iacr.org/2011/135>.  (It's also in one of Dr.
Bernstein's papers, but I don't remember exactly where.)

(But you don't gain anything by doing that for an existing curve.  The
Gaudry et al. formulas are only worth considering if you can choose a
new curve to suit them.)



Robert Ransom