Curve

Documentation

data Coordinates

Curve coordinates.

Constructors

Affine

Jacobian

Projective

data Form

Curve forms.

Constructors

Binary

Edwards

Montgomery

Weierstrass

class (GaloisField q, GaloisField r, Group (Point f c e q r)) => Curve (f :: Form) (c :: Coordinates) e q r where

Elliptic curves.

Minimal complete definition

char, cof, disc, fromA, point, pointX, toA, yX

Associated Types

data Point f c e q r :: *

Curve point.

Methods

char :: Point f c e q r -> Integer

Curve characteristic.

cof :: Point f c e q r -> Integer

Curve cofactor.

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

Curve discriminant.

mul :: r ~ PrimeField p => Point f c e q r -> r -> Point f c e q r

Curve point multiplication.

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

Get point from X and Y coordinates.

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

Get point from X coordinate.

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

Get Y coordinate from X coordinate.

fromA :: Point f Affine e q r -> Point f c e q r

Transform from affine coordinates.

toA :: Point f c e q r -> Point f Affine e q r

Transform to affine coordinates.

Instances

WACurve e q r => Curve Weierstrass Affine (e :: k) q r

Defined in Curve.Weierstrass

Associated Types

data Point Weierstrass Affine e q r :: Type

Methods

char :: Point Weierstrass Affine e q r -> Integer

cof :: Point Weierstrass Affine e q r -> Integer

disc :: Point Weierstrass Affine e q r -> q

mul :: r ~ PrimeField p => Point Weierstrass Affine e q r -> r -> Point Weierstrass Affine e q r

point :: q -> q -> Maybe (Point Weierstrass Affine e q r)

pointX :: q -> Maybe (Point Weierstrass Affine e q r)

yX :: Point Weierstrass Affine e q r -> q -> Maybe q

fromA :: Point Weierstrass Affine e q r -> Point Weierstrass Affine e q r

toA :: Point Weierstrass Affine e q r -> Point Weierstrass Affine e q r

WJCurve e q r => Curve Weierstrass Jacobian (e :: k) q r

Defined in Curve.Weierstrass

Associated Types

data Point Weierstrass Jacobian e q r :: Type

Methods

char :: Point Weierstrass Jacobian e q r -> Integer

cof :: Point Weierstrass Jacobian e q r -> Integer

disc :: Point Weierstrass Jacobian e q r -> q

mul :: r ~ PrimeField p => Point Weierstrass Jacobian e q r -> r -> Point Weierstrass Jacobian e q r

point :: q -> q -> Maybe (Point Weierstrass Jacobian e q r)

pointX :: q -> Maybe (Point Weierstrass Jacobian e q r)

yX :: Point Weierstrass Jacobian e q r -> q -> Maybe q

fromA :: Point Weierstrass Affine e q r -> Point Weierstrass Jacobian e q r

toA :: Point Weierstrass Jacobian e q r -> Point Weierstrass Affine e q r

WPCurve e q r => Curve Weierstrass Projective (e :: k) q r

Defined in Curve.Weierstrass

Associated Types

data Point Weierstrass Projective e q r :: Type

Methods

char :: Point Weierstrass Projective e q r -> Integer

cof :: Point Weierstrass Projective e q r -> Integer

disc :: Point Weierstrass Projective e q r -> q

mul :: r ~ PrimeField p => Point Weierstrass Projective e q r -> r -> Point Weierstrass Projective e q r

point :: q -> q -> Maybe (Point Weierstrass Projective e q r)

pointX :: q -> Maybe (Point Weierstrass Projective e q r)

yX :: Point Weierstrass Projective e q r -> q -> Maybe q

fromA :: Point Weierstrass Affine e q r -> Point Weierstrass Projective e q r

toA :: Point Weierstrass Projective e q r -> Point Weierstrass Affine e q r

MACurve e q r => Curve Montgomery Affine (e :: k) q r

Defined in Curve.Montgomery

Associated Types

data Point Montgomery Affine e q r :: Type

Methods

char :: Point Montgomery Affine e q r -> Integer

cof :: Point Montgomery Affine e q r -> Integer

disc :: Point Montgomery Affine e q r -> q

mul :: r ~ PrimeField p => Point Montgomery Affine e q r -> r -> Point Montgomery Affine e q r

point :: q -> q -> Maybe (Point Montgomery Affine e q r)

pointX :: q -> Maybe (Point Montgomery Affine e q r)

yX :: Point Montgomery Affine e q r -> q -> Maybe q

fromA :: Point Montgomery Affine e q r -> Point Montgomery Affine e q r

toA :: Point Montgomery Affine e q r -> Point Montgomery Affine e q r

EACurve e q r => Curve Edwards Affine (e :: k) q r

Defined in Curve.Edwards

Associated Types

data Point Edwards Affine e q r :: Type

Methods

char :: Point Edwards Affine e q r -> Integer

cof :: Point Edwards Affine e q r -> Integer

disc :: Point Edwards Affine e q r -> q

mul :: r ~ PrimeField p => Point Edwards Affine e q r -> r -> Point Edwards Affine e q r

point :: q -> q -> Maybe (Point Edwards Affine e q r)

pointX :: q -> Maybe (Point Edwards Affine e q r)

yX :: Point Edwards Affine e q r -> q -> Maybe q

fromA :: Point Edwards Affine e q r -> Point Edwards Affine e q r

toA :: Point Edwards Affine e q r -> Point Edwards Affine e q r

EPCurve e q r => Curve Edwards Projective (e :: k) q r

Defined in Curve.Edwards

Associated Types

data Point Edwards Projective e q r :: Type

Methods

char :: Point Edwards Projective e q r -> Integer

cof :: Point Edwards Projective e q r -> Integer

disc :: Point Edwards Projective e q r -> q

mul :: r ~ PrimeField p => Point Edwards Projective e q r -> r -> Point Edwards Projective e q r

point :: q -> q -> Maybe (Point Edwards Projective e q r)

pointX :: q -> Maybe (Point Edwards Projective e q r)

yX :: Point Edwards Projective e q r -> q -> Maybe q

fromA :: Point Edwards Affine e q r -> Point Edwards Projective e q r

toA :: Point Edwards Projective e q r -> Point Edwards Affine e q r

BACurve e q r => Curve Binary Affine (e :: k) q r

Defined in Curve.Binary

Associated Types

data Point Binary Affine e q r :: Type

Methods

char :: Point Binary Affine e q r -> Integer

cof :: Point Binary Affine e q r -> Integer

disc :: Point Binary Affine e q r -> q

mul :: r ~ PrimeField p => Point Binary Affine e q r -> r -> Point Binary Affine e q r

point :: q -> q -> Maybe (Point Binary Affine e q r)

pointX :: q -> Maybe (Point Binary Affine e q r)

yX :: Point Binary Affine e q r -> q -> Maybe q

fromA :: Point Binary Affine e q r -> Point Binary Affine e q r

toA :: Point Binary Affine e q r -> Point Binary Affine e q r

BPCurve e q r => Curve Binary Projective (e :: k) q r

Defined in Curve.Binary

Associated Types

data Point Binary Projective e q r :: Type

Methods

char :: Point Binary Projective e q r -> Integer

cof :: Point Binary Projective e q r -> Integer

disc :: Point Binary Projective e q r -> q

mul :: r ~ PrimeField p => Point Binary Projective e q r -> r -> Point Binary Projective e q r

point :: q -> q -> Maybe (Point Binary Projective e q r)

pointX :: q -> Maybe (Point Binary Projective e q r)

yX :: Point Binary Projective e q r -> q -> Maybe q

fromA :: Point Binary Affine e q r -> Point Binary Projective e q r

toA :: Point Binary Projective e q r -> Point Binary Affine e q r

class (Arbitrary g, Eq g, Generic g, Monoid g, Pretty g, Random g, Show g) => Group g where

Groups.

Minimal complete definition

add, dbl, def, gen, id, inv, order

Methods

add :: g -> g -> g

Element addition.

dbl :: g -> g

Element doubling.

def :: g -> Bool

Check well-defined.

gen :: g

Group generator.

id :: g

Identity element.

inv :: g -> g

Element inversion.

mul' :: g -> Integer -> g

Element multiplication.

order :: g -> Integer

Curve order.

rnd :: MonadRandom m => m g

Random element.

Instances

FGroup k => Group (Element k)

Defined in Group.Field

Methods

add :: Element k -> Element k -> Element k

dbl :: Element k -> Element k

def :: Element k -> Bool

gen :: Element k

id :: Element k

inv :: Element k -> Element k

mul' :: Element k -> Integer -> Element k

order :: Element k -> Integer

rnd :: MonadRandom m => m (Element k)

WPCurve e q r => Group (WPPoint e q r)

Defined in Curve.Weierstrass

Methods

add :: WPPoint e q r -> WPPoint e q r -> WPPoint e q r

dbl :: WPPoint e q r -> WPPoint e q r

def :: WPPoint e q r -> Bool

gen :: WPPoint e q r

id :: WPPoint e q r

inv :: WPPoint e q r -> WPPoint e q r

mul' :: WPPoint e q r -> Integer -> WPPoint e q r

order :: WPPoint e q r -> Integer

rnd :: MonadRandom m => m (WPPoint e q r)

WJCurve e q r => Group (WJPoint e q r)

Defined in Curve.Weierstrass

Methods

add :: WJPoint e q r -> WJPoint e q r -> WJPoint e q r

dbl :: WJPoint e q r -> WJPoint e q r

def :: WJPoint e q r -> Bool

gen :: WJPoint e q r

id :: WJPoint e q r

inv :: WJPoint e q r -> WJPoint e q r

mul' :: WJPoint e q r -> Integer -> WJPoint e q r

order :: WJPoint e q r -> Integer

rnd :: MonadRandom m => m (WJPoint e q r)

WACurve e q r => Group (WAPoint e q r)

Defined in Curve.Weierstrass

Methods

add :: WAPoint e q r -> WAPoint e q r -> WAPoint e q r

dbl :: WAPoint e q r -> WAPoint e q r

def :: WAPoint e q r -> Bool

gen :: WAPoint e q r

id :: WAPoint e q r

inv :: WAPoint e q r -> WAPoint e q r

mul' :: WAPoint e q r -> Integer -> WAPoint e q r

order :: WAPoint e q r -> Integer

rnd :: MonadRandom m => m (WAPoint e q r)

MACurve e q r => Group (MAPoint e q r)

Defined in Curve.Montgomery

Methods

add :: MAPoint e q r -> MAPoint e q r -> MAPoint e q r

dbl :: MAPoint e q r -> MAPoint e q r

def :: MAPoint e q r -> Bool

gen :: MAPoint e q r

id :: MAPoint e q r

inv :: MAPoint e q r -> MAPoint e q r

mul' :: MAPoint e q r -> Integer -> MAPoint e q r

order :: MAPoint e q r -> Integer

rnd :: MonadRandom m => m (MAPoint e q r)

EPCurve e q r => Group (EPPoint e q r)

Defined in Curve.Edwards

Methods

add :: EPPoint e q r -> EPPoint e q r -> EPPoint e q r

dbl :: EPPoint e q r -> EPPoint e q r

def :: EPPoint e q r -> Bool

gen :: EPPoint e q r

id :: EPPoint e q r

inv :: EPPoint e q r -> EPPoint e q r

mul' :: EPPoint e q r -> Integer -> EPPoint e q r

order :: EPPoint e q r -> Integer

rnd :: MonadRandom m => m (EPPoint e q r)

EACurve e q r => Group (EAPoint e q r)

Defined in Curve.Edwards

Methods

add :: EAPoint e q r -> EAPoint e q r -> EAPoint e q r

dbl :: EAPoint e q r -> EAPoint e q r

def :: EAPoint e q r -> Bool

gen :: EAPoint e q r

id :: EAPoint e q r

inv :: EAPoint e q r -> EAPoint e q r

mul' :: EAPoint e q r -> Integer -> EAPoint e q r

order :: EAPoint e q r -> Integer

rnd :: MonadRandom m => m (EAPoint e q r)

BPCurve e q r => Group (BPPoint e q r)

Defined in Curve.Binary

Methods

add :: BPPoint e q r -> BPPoint e q r -> BPPoint e q r

dbl :: BPPoint e q r -> BPPoint e q r

def :: BPPoint e q r -> Bool

gen :: BPPoint e q r

id :: BPPoint e q r

inv :: BPPoint e q r -> BPPoint e q r

mul' :: BPPoint e q r -> Integer -> BPPoint e q r

order :: BPPoint e q r -> Integer

rnd :: MonadRandom m => m (BPPoint e q r)

BACurve e q r => Group (BAPoint e q r)

Defined in Curve.Binary

Methods

add :: BAPoint e q r -> BAPoint e q r -> BAPoint e q r

dbl :: BAPoint e q r -> BAPoint e q r

def :: BAPoint e q r -> Bool

gen :: BAPoint e q r

id :: BAPoint e q r

inv :: BAPoint e q r -> BAPoint e q r

mul' :: BAPoint e q r -> Integer -> BAPoint e q r

order :: BAPoint e q r -> Integer

rnd :: MonadRandom m => m (BAPoint e q r)