[curves] Isogeny patterns among Edwards curves

Robert Ransom rransom.8774 at gmail.com
Wed Jan 29 18:54:10 PST 2014

On 1/29/14, Samuel Neves <sneves at dei.uc.pt> wrote:
> On 29-01-2014 18:22, Mike Hamburg wrote:
>>> The second pattern is that Mont(1, 4*d + 2) is isogenous to Ed(-1,
>>> d).  For example:
>>> I noticed this pattern while looking for a twist-secure
>>> small-parameter Edwards curve over the Curve25519 coordinate field:
>>> the first winning curve had a value of d suspiciously similar to
>>> (A+2)/4 for one of the curves that Dr. Bernstein considered in the
>>> Curve25519 paper.  Further experiments showed that the pattern held
>>> for the other two curves considered there, including Curve25519
>>> itself.
>> This is neat, and I didn't know it before you mentioned it on the
>> Montgomery thread.  Do you know what the isogeny is?  Is it simple,
>> and does it interact nicely with point compression?  Is it a 2-isogeny
>> or a 4-isogeny?

> It can be derived from [1]. Composing E_{1, d} -> E_{-1, d-1} with E_d
> -> E_{1-1/d} (Section 3), and using the identity A = 4/(1 - d) - 2, one
> gets the isogeny E_{-1, d} -> M_{4*d + 2}. From what I can tell this is
> a 4-isogeny.
> [1] http://eprint.iacr.org/2011/135

Thank you!

Robert Ransom

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