# Twisted Edwards Curves

Twisted Edwards curves are parameterized by $$a, d$$ and are of the form

$$
\mathcal E\_{a,d}​:ax^2+y^2=1+dx^2y^2.
$$

These are usually represented by the [Extended Twisted Edwards Coordinates](https://eprint.iacr.org/2008/522.pdf) of Hisil, Wong, Carter, and Dawson: points are represented in projective coordinates as $$(X : Y : Z : T)$$ with

$$
XY=ZT,\ \ aX^2+Y^2=Z^2+dT^2.
$$

(More details on Edwards curve models can be found in the [`curve25519_dalek` `curve_models`](https://doc-internal.dalek.rs/curve25519_dalek/curve_models/index.html) documentation). The case $$a = 1$$ is the *untwisted* case; the case $$a = -1$$ provides the fastest formulas. When not otherwise specified, we $$\mathcal E$$ for $$\mathcal E\_{a,d}$$ ​.

When both $$d$$ and $$ad$$ are nonsquare (which forces $$a$$ to be square), the curve is *complete*. In this case the four-torsion subgroup is cyclic, and we can write it explicitly as

$$
\mathcal E\_{a,d}​\[4] = {(0,1), (1/\sqrt a, 0), (0, -1), (-1/\sqrt a, 0)}
$$

These are the only points with $$xy = 0$$ ; the points with $$y \neq 0$$ are 2-torsion.


---

# Agent Instructions: Querying This Documentation

If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://wiki.corecoin.cc/wiki/cryptography/twisted-edwards-curves.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.