# Elliptic Curve

An extensible library of elliptic curves used in cryptography research.

## Curve representations

An **elliptic curve** E(K) over a field K is a *smooth projective plane algebraic cubic curve* with a specified base point `O`

, and the *points* on E(K) form an *algebraic group* with identity point `O`

. By the *Riemann-Roch theorem*, any elliptic curve is isomorphic to a cubic curve of the form

where are *K-rational coefficients* that satisfy a *non-zero discriminant* condition. For cryptographic computational purposes, elliptic curves are represented in several different forms.

### Weierstrass curves

A (short) , and is of the form

where `A`

and `B`

are K-rational coefficients such that is non-zero. Weierstrass curves are the most common representations of elliptic curves, as any elliptic curve over a field of characteristic greater than 3 is isomorphic to a Weierstrass curve.

### Binary curves

A (short Weierstrass) , and is of the form

where `A`

and `B`

are K-rational coefficients such that `B`

is non-zero. Binary curves have field elements represented by binary integers for efficient arithmetic, and are special cases of *long Weierstrass curves* over a field of characteristic 2.

### Montgomery curves

where `A`

and `B`

are K-rational coefficients such that is non-zero. Montgomery curves only use the first affine coordinate for computations, and can utilise the Montgomery ladder for efficient multiplication.

### Edwards curves

A (twisted) , and is of the form

where `A`

and `D`

are K-rational coefficients such that is non-zero. Edwards curves have no point at infinity, and their addition and doubling formulae converge.

## Curve usage

This library is open for new curve representations and curve implementations through pull requests. These should ideally be executed by replicating and modifying existing curve files, for ease, quickcheck testing, and formatting consistency, but a short description of the file organisation is provided here for clarity. Note that it also has a dependency on the Galois field library and its required language extensions.

The library exposes four promoted data kinds which are used to define a type-safe interface for working with curves.

**Forms**

**Coordinates**

These are then specialised down into type classes for the different forms.

And then by coordinate system.

A curve class is constructed out of four type parameters which are instantiated in the associated data type Point on the Curve typeclass.

```
class Curve (f :: Form) (c :: Coordinates) e q r
| | |
Curve Type o-+ | |
Field of Points o---+ |
Field of Coefficients o-----+
data Point f c e q r :: *
```

For example:

```
data Anomalous
type Fq = Prime Q
type Q = 0xb0000000000000000000000953000000000000000000001f9d7
type Fr = Prime R
type R = 0xb0000000000000000000000953000000000000000000001f9d7
instance Curve 'Weierstrass c Anomalous Fq Fr => WCurve c Anomalous Fq Fr where
-- data instance Point 'Weierstrass c Anomalous Fq Fr
```

**Arithmetic**

```
-- Point addition
add :: Point f c e q r -> Point f c e q r -> Point f c e q r
-- Point doubling
dbl :: Point f c e q r -> Point f c e q r
-- Point multiplication by field element
mul :: Curve f c e q r => Point f c e q r -> r -> Point f c e q r
-- Point multiplication by Integral
mul' :: (Curve f c e q r, Integral n) => Point f c e q r -> n -> Point f c e q r
-- Point identity
id :: Point f c e q r
-- Point inversion
inv :: Point f c e q r -> Point f c e q r
-- Frobenius endomorphism
frob :: Point f c e q r -> Point f c e q r
-- Random point
rnd :: MonadRandom m => m (Point f c e q r)
```

**Other Functions**

```
-- Curve characteristic
char :: Point f c e q r -> Natural
-- Curve cofactor
cof :: Point f c e q r -> Natural
-- Check if a point is well-defined
def :: Point f c e q r -> Bool
-- Discriminant
disc :: Point f c e q r -> q
-- Curve order
order :: Point f c e q r -> Natural
-- Curve generator point
gen :: Point f c e q r
```

### Point Arithmetic

See **Example.hs**.

### Elliptic Curve Diffie-Hellman (ECDH)

See **DiffieHellman.hs**.

### Representing a new curve using the curve class

See **Weierstrass**.

### Implementing a new curve using a curve representation

See **Anomalous**.

### Using an implemented curve

Import a curve implementation.

`import qualified Data.Curve.Weierstrass.Anomalous as Anomalous`

The data types and constants can then be accessed readily as `Anomalous.PA`

and `Anomalous._g`

.

We'll test that the Hasse Theorem is successful with an implemented curve as a usage example:

```
import Protolude
import GHC.Natural
import qualified Data.Field.Galois as F
main :: IO ()
main = do
putText $ "Hasse Theorem succeeds: " <> show (hasseTheorem Anomalous._h Anomalous._r (F.order (witness :: Anomalous.Fq)))
where
hasseTheorem h r q = join (*) (naturalToInteger (h * r) - naturalToInteger q - 1) <= 4 * naturalToInteger q
```

## Curve implementations

The following curves have already been implemented.

### Binary curves

- SECT (NIST) curves

### Edwards curves

- Edwards curves

### Montgomery curves

- Montgomery curves

### Weierstrass curves

- Anomalous
- ANSSIFRP256V1
- Barreto-Lynn-Scott (BLS) curves
- Barreto-Naehrig (BN) curves
- Brainpool curves
- SECP (NIST) curves

## Disclaimer

The data structures in this library are meant for use in research-grade projects
and not in interactive protocols. The elliptic curve operations in this library
are *not constant time* and thus may be vulernable to timing attacks if used
improperly. If you are unsure of the implications of this, *do not use this
library*.

## License

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