Birational transformation from Edwards curve with not square d to Edwards curve with square d

ng flag

How can I transform a complete twisted Edwards curve $ax^2+y^2 = 1+dx^2y^2$ with not square $d$ and square $a$ into an isomorphic Edwards curve $X^2+Y^2 = 1+DX^2Y^2$ with a square $-D$ i.e. $D = -r^2$?

I tried to set $X = \frac{x}{\sqrt{a}}; Y=y$, but $-\frac{d}{a}$ is also a non square (at least for Edwards25519). This answer is not working as well (i.e. $-1/d$ is not a square), because $-1$ is square.

Is it even possible to do such a transformation?

Fractalice avatar
in flag
If your number $d$ is not a square, you can always work in the extension field, where the square does exist. E.g. work in $F[u]/(u^2-d)$, where $F$ is your original field.
pintor avatar
ng flag
@Fractalice, Thanks! However, I'm not sure it will work for the injective encoding I'm implementing ( Is there any other way? Btw, how to find $u$? Is it a square root of something?

Post an answer

Most people don’t grasp that asking a lot of questions unlocks learning and improves interpersonal bonding. In Alison’s studies, for example, though people could accurately recall how many questions had been asked in their conversations, they didn’t intuit the link between questions and liking. Across four studies, in which participants were engaged in conversations themselves or read transcripts of others’ conversations, people tended not to realize that question asking would influence—or had influenced—the level of amity between the conversationalists.