Safe Haskell	None
Language	Haskell2010

Data.Curve

Contents

Elliptic curves
- Elliptic curve forms
- Elliptic curve coordinates

Synopsis

class (GaloisField q, PrimeField r, Arbitrary (Point f c e q r), Eq (Point f c e q r), Generic (Point f c e q r), Group (Point f c e q r), NFData (Point f c e q r), Pretty (Point f c e q r), Random (Point f c e q r), Show (Point f c e q r)) => Curve (f :: Form) (c :: Coordinates) e q r where
- data Point f c e q r :: *
- char :: Point f c e q r -> Natural
- cof :: Point f c e q r -> Natural
- def :: Point f c e q r -> Bool
- disc :: Point f c e q r -> q
- order :: Point f c e q r -> Natural
- add :: Point f c e q r -> Point f c e q r -> Point f c e q r
- dbl :: Point f c e q r -> Point f c e q r
- id :: Point f c e q r
- inv :: Point f c e q r -> Point f c e q r
- frob :: Point f c e q r -> Point f c e q r
- fromA :: Curve f Affine e q r => Point f Affine e q r -> Point f c e q r
- gen :: Point f c e q r
- point :: q -> q -> Maybe (Point f c e q r)
- pointX :: q -> Maybe (Point f c e q r)
- rnd :: MonadRandom m => m (Point f c e q r)
- toA :: Curve f Affine e q r => Point f c e q r -> Point f Affine e q r
- yX :: Point f c e q r -> q -> Maybe q
mul :: Curve f c e q r => Point f c e q r -> r -> Point f c e q r
mul' :: (Curve f c e q r, Integral n) => Point f c e q r -> n -> Point f c e q r
data Form
- = Binary
- | Edwards
- | Montgomery
- | Weierstrass
data Coordinates
- = Affine
- | Jacobian
- | Projective

Elliptic curves

class (GaloisField q, PrimeField r, Arbitrary (Point f c e q r), Eq (Point f c e q r), Generic (Point f c e q r), Group (Point f c e q r), NFData (Point f c e q r), Pretty (Point f c e q r), Random (Point f c e q r), Show (Point f c e q r)) => Curve (f :: Form) (c :: Coordinates) e q r where Source #

Elliptic curves.

Minimal complete definition

add, char, cof, dbl, def, disc, frob, fromA, gen, id, inv, order, point, pointX, toA, yX

Associated Types

data Point f c e q r :: * Source #

Curve point.

Methods

char :: Point f c e q r -> Natural Source #

Curve characteristic.

cof :: Point f c e q r -> Natural Source #

Curve cofactor.

def :: Point f c e q r -> Bool Source #

Curve well-defined.

disc :: Point f c e q r -> q Source #

Curve discriminant.

order :: Point f c e q r -> Natural Source #

Curve order.

add :: Point f c e q r -> Point f c e q r -> Point f c e q r Source #

Point addition.

dbl :: Point f c e q r -> Point f c e q r Source #

Point doubling.

id :: Point f c e q r Source #

Point identity.

inv :: Point f c e q r -> Point f c e q r Source #

Point inversion.

frob :: Point f c e q r -> Point f c e q r Source #

Frobenius endomorphism.

fromA :: Curve f Affine e q r => Point f Affine e q r -> Point f c e q r Source #

Transform from affine coordinates.

gen :: Point f c e q r Source #

Curve generator.

point :: q -> q -> Maybe (Point f c e q r) Source #

Get point from X and Y coordinates.

pointX :: q -> Maybe (Point f c e q r) Source #

Get point from X coordinate.

rnd :: MonadRandom m => m (Point f c e q r) Source #

Random point.

toA :: Curve f Affine e q r => Point f c e q r -> Point f Affine e q r Source #

Transform to affine coordinates.

yX :: Point f c e q r -> q -> Maybe q Source #

Get Y coordinate from X coordinate.

Instances

BACurve e q r => Curve Binary Affine (e :: k) q r Source #
Instance details Defined in Data.Curve.Binary Associated Types data Point Binary Affine e q r :: Type Source # Methods char :: Point Binary Affine e q r -> Natural Source # cof :: Point Binary Affine e q r -> Natural Source # def :: Point Binary Affine e q r -> Bool Source # disc :: Point Binary Affine e q r -> q Source # order :: Point Binary Affine e q r -> Natural Source # add :: Point Binary Affine e q r -> Point Binary Affine e q r -> Point Binary Affine e q r Source # dbl :: Point Binary Affine e q r -> Point Binary Affine e q r Source # id :: Point Binary Affine e q r Source # inv :: Point Binary Affine e q r -> Point Binary Affine e q r Source # frob :: Point Binary Affine e q r -> Point Binary Affine e q r Source # fromA :: Point Binary Affine e q r -> Point Binary Affine e q r Source # gen :: Point Binary Affine e q r Source # point :: q -> q -> Maybe (Point Binary Affine e q r) Source # pointX :: q -> Maybe (Point Binary Affine e q r) Source # rnd :: MonadRandom m => m (Point Binary Affine e q r) Source # toA :: Point Binary Affine e q r -> Point Binary Affine e q r Source # yX :: Point Binary Affine e q r -> q -> Maybe q Source #
BPCurve e q r => Curve Binary Projective (e :: k) q r Source #
Instance details Defined in Data.Curve.Binary Associated Types data Point Binary Projective e q r :: Type Source # Methods char :: Point Binary Projective e q r -> Natural Source # cof :: Point Binary Projective e q r -> Natural Source # def :: Point Binary Projective e q r -> Bool Source # disc :: Point Binary Projective e q r -> q Source # order :: Point Binary Projective e q r -> Natural Source # add :: Point Binary Projective e q r -> Point Binary Projective e q r -> Point Binary Projective e q r Source # dbl :: Point Binary Projective e q r -> Point Binary Projective e q r Source # id :: Point Binary Projective e q r Source # inv :: Point Binary Projective e q r -> Point Binary Projective e q r Source # frob :: Point Binary Projective e q r -> Point Binary Projective e q r Source # fromA :: Point Binary Affine e q r -> Point Binary Projective e q r Source # gen :: Point Binary Projective e q r Source # point :: q -> q -> Maybe (Point Binary Projective e q r) Source # pointX :: q -> Maybe (Point Binary Projective e q r) Source # rnd :: MonadRandom m => m (Point Binary Projective e q r) Source # toA :: Point Binary Projective e q r -> Point Binary Affine e q r Source # yX :: Point Binary Projective e q r -> q -> Maybe q Source #
EACurve e q r => Curve Edwards Affine (e :: k) q r Source #
Instance details Defined in Data.Curve.Edwards Associated Types data Point Edwards Affine e q r :: Type Source # Methods char :: Point Edwards Affine e q r -> Natural Source # cof :: Point Edwards Affine e q r -> Natural Source # def :: Point Edwards Affine e q r -> Bool Source # disc :: Point Edwards Affine e q r -> q Source # order :: Point Edwards Affine e q r -> Natural Source # add :: Point Edwards Affine e q r -> Point Edwards Affine e q r -> Point Edwards Affine e q r Source # dbl :: Point Edwards Affine e q r -> Point Edwards Affine e q r Source # id :: Point Edwards Affine e q r Source # inv :: Point Edwards Affine e q r -> Point Edwards Affine e q r Source # frob :: Point Edwards Affine e q r -> Point Edwards Affine e q r Source # fromA :: Point Edwards Affine e q r -> Point Edwards Affine e q r Source # gen :: Point Edwards Affine e q r Source # point :: q -> q -> Maybe (Point Edwards Affine e q r) Source # pointX :: q -> Maybe (Point Edwards Affine e q r) Source # rnd :: MonadRandom m => m (Point Edwards Affine e q r) Source # toA :: Point Edwards Affine e q r -> Point Edwards Affine e q r Source # yX :: Point Edwards Affine e q r -> q -> Maybe q Source #
EPCurve e q r => Curve Edwards Projective (e :: k) q r Source #
Instance details Defined in Data.Curve.Edwards Associated Types data Point Edwards Projective e q r :: Type Source # Methods char :: Point Edwards Projective e q r -> Natural Source # cof :: Point Edwards Projective e q r -> Natural Source # def :: Point Edwards Projective e q r -> Bool Source # disc :: Point Edwards Projective e q r -> q Source # order :: Point Edwards Projective e q r -> Natural Source # add :: Point Edwards Projective e q r -> Point Edwards Projective e q r -> Point Edwards Projective e q r Source # dbl :: Point Edwards Projective e q r -> Point Edwards Projective e q r Source # id :: Point Edwards Projective e q r Source # inv :: Point Edwards Projective e q r -> Point Edwards Projective e q r Source # frob :: Point Edwards Projective e q r -> Point Edwards Projective e q r Source # fromA :: Point Edwards Affine e q r -> Point Edwards Projective e q r Source # gen :: Point Edwards Projective e q r Source # point :: q -> q -> Maybe (Point Edwards Projective e q r) Source # pointX :: q -> Maybe (Point Edwards Projective e q r) Source # rnd :: MonadRandom m => m (Point Edwards Projective e q r) Source # toA :: Point Edwards Projective e q r -> Point Edwards Affine e q r Source # yX :: Point Edwards Projective e q r -> q -> Maybe q Source #
MACurve e q r => Curve Montgomery Affine (e :: k) q r Source #
Instance details Defined in Data.Curve.Montgomery Associated Types data Point Montgomery Affine e q r :: Type Source # Methods char :: Point Montgomery Affine e q r -> Natural Source # cof :: Point Montgomery Affine e q r -> Natural Source # def :: Point Montgomery Affine e q r -> Bool Source # disc :: Point Montgomery Affine e q r -> q Source # order :: Point Montgomery Affine e q r -> Natural Source # add :: Point Montgomery Affine e q r -> Point Montgomery Affine e q r -> Point Montgomery Affine e q r Source # dbl :: Point Montgomery Affine e q r -> Point Montgomery Affine e q r Source # id :: Point Montgomery Affine e q r Source # inv :: Point Montgomery Affine e q r -> Point Montgomery Affine e q r Source # frob :: Point Montgomery Affine e q r -> Point Montgomery Affine e q r Source # fromA :: Point Montgomery Affine e q r -> Point Montgomery Affine e q r Source # gen :: Point Montgomery Affine e q r Source # point :: q -> q -> Maybe (Point Montgomery Affine e q r) Source # pointX :: q -> Maybe (Point Montgomery Affine e q r) Source # rnd :: MonadRandom m => m (Point Montgomery Affine e q r) Source # toA :: Point Montgomery Affine e q r -> Point Montgomery Affine e q r Source # yX :: Point Montgomery Affine e q r -> q -> Maybe q Source #
WACurve e q r => Curve Weierstrass Affine (e :: k) q r Source #
Instance details Defined in Data.Curve.Weierstrass Associated Types data Point Weierstrass Affine e q r :: Type Source # Methods char :: Point Weierstrass Affine e q r -> Natural Source # cof :: Point Weierstrass Affine e q r -> Natural Source # def :: Point Weierstrass Affine e q r -> Bool Source # disc :: Point Weierstrass Affine e q r -> q Source # order :: Point Weierstrass Affine e q r -> Natural Source # add :: Point Weierstrass Affine e q r -> Point Weierstrass Affine e q r -> Point Weierstrass Affine e q r Source # dbl :: Point Weierstrass Affine e q r -> Point Weierstrass Affine e q r Source # id :: Point Weierstrass Affine e q r Source # inv :: Point Weierstrass Affine e q r -> Point Weierstrass Affine e q r Source # frob :: Point Weierstrass Affine e q r -> Point Weierstrass Affine e q r Source # fromA :: Point Weierstrass Affine e q r -> Point Weierstrass Affine e q r Source # gen :: Point Weierstrass Affine e q r Source # point :: q -> q -> Maybe (Point Weierstrass Affine e q r) Source # pointX :: q -> Maybe (Point Weierstrass Affine e q r) Source # rnd :: MonadRandom m => m (Point Weierstrass Affine e q r) Source # toA :: Point Weierstrass Affine e q r -> Point Weierstrass Affine e q r Source # yX :: Point Weierstrass Affine e q r -> q -> Maybe q Source #
WJCurve e q r => Curve Weierstrass Jacobian (e :: k) q r Source #
Instance details Defined in Data.Curve.Weierstrass Associated Types data Point Weierstrass Jacobian e q r :: Type Source # Methods char :: Point Weierstrass Jacobian e q r -> Natural Source # cof :: Point Weierstrass Jacobian e q r -> Natural Source # def :: Point Weierstrass Jacobian e q r -> Bool Source # disc :: Point Weierstrass Jacobian e q r -> q Source # order :: Point Weierstrass Jacobian e q r -> Natural Source # add :: Point Weierstrass Jacobian e q r -> Point Weierstrass Jacobian e q r -> Point Weierstrass Jacobian e q r Source # dbl :: Point Weierstrass Jacobian e q r -> Point Weierstrass Jacobian e q r Source # id :: Point Weierstrass Jacobian e q r Source # inv :: Point Weierstrass Jacobian e q r -> Point Weierstrass Jacobian e q r Source # frob :: Point Weierstrass Jacobian e q r -> Point Weierstrass Jacobian e q r Source # fromA :: Point Weierstrass Affine e q r -> Point Weierstrass Jacobian e q r Source # gen :: Point Weierstrass Jacobian e q r Source # point :: q -> q -> Maybe (Point Weierstrass Jacobian e q r) Source # pointX :: q -> Maybe (Point Weierstrass Jacobian e q r) Source # rnd :: MonadRandom m => m (Point Weierstrass Jacobian e q r) Source # toA :: Point Weierstrass Jacobian e q r -> Point Weierstrass Affine e q r Source # yX :: Point Weierstrass Jacobian e q r -> q -> Maybe q Source #
WPCurve e q r => Curve Weierstrass Projective (e :: k) q r Source #
Instance details Defined in Data.Curve.Weierstrass Associated Types data Point Weierstrass Projective e q r :: Type Source # Methods char :: Point Weierstrass Projective e q r -> Natural Source # cof :: Point Weierstrass Projective e q r -> Natural Source # def :: Point Weierstrass Projective e q r -> Bool Source # disc :: Point Weierstrass Projective e q r -> q Source # order :: Point Weierstrass Projective e q r -> Natural Source # add :: Point Weierstrass Projective e q r -> Point Weierstrass Projective e q r -> Point Weierstrass Projective e q r Source # dbl :: Point Weierstrass Projective e q r -> Point Weierstrass Projective e q r Source # id :: Point Weierstrass Projective e q r Source # inv :: Point Weierstrass Projective e q r -> Point Weierstrass Projective e q r Source # frob :: Point Weierstrass Projective e q r -> Point Weierstrass Projective e q r Source # fromA :: Point Weierstrass Affine e q r -> Point Weierstrass Projective e q r Source # gen :: Point Weierstrass Projective e q r Source # point :: q -> q -> Maybe (Point Weierstrass Projective e q r) Source # pointX :: q -> Maybe (Point Weierstrass Projective e q r) Source # rnd :: MonadRandom m => m (Point Weierstrass Projective e q r) Source # toA :: Point Weierstrass Projective e q r -> Point Weierstrass Affine e q r Source # yX :: Point Weierstrass Projective e q r -> q -> Maybe q Source #

mul :: Curve f c e q r => Point f c e q r -> r -> Point f c e q r Source #

Point multiplication by field element.

mul' :: (Curve f c e q r, Integral n) => Point f c e q r -> n -> Point f c e q r Source #

Point multiplication by integral element.

Elliptic curve forms

data Form Source #

Curve forms.

Constructors

Binary
Edwards
Montgomery
Weierstrass

Elliptic curve coordinates

data Coordinates Source #

Curve coordinates.

Constructors

Affine
Jacobian
Projective