Curve.Montgomery

Documentation

class GaloisField k => PrimeField' k

Minimal complete definition

toInt'

Instances

KnownNat p => PrimeField' (PrimeField p)

Defined in Curve

Methods

toInt' :: PrimeField p -> Integer

data Coordinates

Curve coordinates.

Constructors

Affine

Jacobian

Projective

data Form

Curve forms.

Constructors

Binary

Edwards

Montgomery

Weierstrass

class (GaloisField q, PrimeField' 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 :: 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 :: 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 :: 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 :: 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 :: 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 :: 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 :: 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 :: 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 :: 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 MCurve Affine e q r => MACurve e q r where

Montgomery affine curves By^2 = x^3 + Ax^2 + x.

Methods

gA_

Arguments

:: MAPoint e q r

Curve generator.

Instances

MACurve M511 Fq Fr

Affine M511 curve is a Montgomery affine curve.

Defined in Curve.Montgomery.M511

Methods

gA_ :: MAPoint M511 Fq Fr

MACurve M383 Fq Fr

Affine M383 curve is a Montgomery affine curve.

Defined in Curve.Montgomery.M383

Methods

gA_ :: MAPoint M383 Fq Fr

MACurve M221 Fq Fr

Affine M221 curve is a Montgomery affine curve.

Defined in Curve.Montgomery.M221

Methods

gA_ :: MAPoint M221 Fq Fr

MACurve Curve448 Fq Fr

Affine Curve448 curve is a Montgomery affine curve.

Defined in Curve.Montgomery.Curve448

Methods

gA_ :: MAPoint Curve448 Fq Fr

MACurve Curve383187 Fq Fr

Affine Curve383187 curve is a Montgomery affine curve.

Defined in Curve.Montgomery.Curve383187

Methods

gA_ :: MAPoint Curve383187 Fq Fr

MACurve Curve25519 Fq Fr

Affine Curve25519 curve is a Montgomery affine curve.

Defined in Curve.Montgomery.Curve25519

Methods

gA_ :: MAPoint Curve25519 Fq Fr

type MAPoint = MPoint Affine

Montgomery affine points.

class (GaloisField q, PrimeField' r, Curve Montgomery c e q r) => MCurve c e q r where

Montgomery curves.

Methods

a_

Arguments

:: MPoint c e q r

-> q

Coefficient A.

b_

Arguments

:: MPoint c e q r

-> q

Coefficient B.

h_

Arguments

:: MPoint c e q r

-> Integer

Curve cofactor.

q_

Arguments

:: MPoint c e q r

-> Integer

Curve characteristic.

r_

Arguments

:: MPoint c e q r

-> Integer

Curve order.

x_

Arguments

:: MPoint c e q r

-> q

Coordinate X.

y_

Arguments

:: MPoint c e q r

-> q

Coordinate Y.

Instances

Curve Montgomery c M511 Fq Fr => MCurve c M511 Fq Fr

M511 curve is a Montgomery curve.

Defined in Curve.Montgomery.M511

Methods

a_ :: MPoint c M511 Fq Fr -> Fq

b_ :: MPoint c M511 Fq Fr -> Fq

h_ :: MPoint c M511 Fq Fr -> Integer

q_ :: MPoint c M511 Fq Fr -> Integer

r_ :: MPoint c M511 Fq Fr -> Integer

x_ :: MPoint c M511 Fq Fr -> Fq

y_ :: MPoint c M511 Fq Fr -> Fq

Curve Montgomery c M383 Fq Fr => MCurve c M383 Fq Fr

M383 curve is a Montgomery curve.

Defined in Curve.Montgomery.M383

Methods

a_ :: MPoint c M383 Fq Fr -> Fq

b_ :: MPoint c M383 Fq Fr -> Fq

h_ :: MPoint c M383 Fq Fr -> Integer

q_ :: MPoint c M383 Fq Fr -> Integer

r_ :: MPoint c M383 Fq Fr -> Integer

x_ :: MPoint c M383 Fq Fr -> Fq

y_ :: MPoint c M383 Fq Fr -> Fq

Curve Montgomery c M221 Fq Fr => MCurve c M221 Fq Fr

M221 curve is a Montgomery curve.

Defined in Curve.Montgomery.M221

Methods

a_ :: MPoint c M221 Fq Fr -> Fq

b_ :: MPoint c M221 Fq Fr -> Fq

h_ :: MPoint c M221 Fq Fr -> Integer

q_ :: MPoint c M221 Fq Fr -> Integer

r_ :: MPoint c M221 Fq Fr -> Integer

x_ :: MPoint c M221 Fq Fr -> Fq

y_ :: MPoint c M221 Fq Fr -> Fq

Curve Montgomery c Curve448 Fq Fr => MCurve c Curve448 Fq Fr

Curve448 curve is a Montgomery curve.

Defined in Curve.Montgomery.Curve448

Methods

a_ :: MPoint c Curve448 Fq Fr -> Fq

b_ :: MPoint c Curve448 Fq Fr -> Fq

h_ :: MPoint c Curve448 Fq Fr -> Integer

q_ :: MPoint c Curve448 Fq Fr -> Integer

r_ :: MPoint c Curve448 Fq Fr -> Integer

x_ :: MPoint c Curve448 Fq Fr -> Fq

y_ :: MPoint c Curve448 Fq Fr -> Fq

Curve Montgomery c Curve383187 Fq Fr => MCurve c Curve383187 Fq Fr

Curve383187 curve is a Montgomery curve.

Defined in Curve.Montgomery.Curve383187

Methods

a_ :: MPoint c Curve383187 Fq Fr -> Fq

b_ :: MPoint c Curve383187 Fq Fr -> Fq

h_ :: MPoint c Curve383187 Fq Fr -> Integer

q_ :: MPoint c Curve383187 Fq Fr -> Integer

r_ :: MPoint c Curve383187 Fq Fr -> Integer

x_ :: MPoint c Curve383187 Fq Fr -> Fq

y_ :: MPoint c Curve383187 Fq Fr -> Fq

Curve Montgomery c Curve25519 Fq Fr => MCurve c Curve25519 Fq Fr

Curve25519 curve is a Montgomery curve.

Defined in Curve.Montgomery.Curve25519

Methods

a_ :: MPoint c Curve25519 Fq Fr -> Fq

b_ :: MPoint c Curve25519 Fq Fr -> Fq

h_ :: MPoint c Curve25519 Fq Fr -> Integer

q_ :: MPoint c Curve25519 Fq Fr -> Integer

r_ :: MPoint c Curve25519 Fq Fr -> Integer

x_ :: MPoint c Curve25519 Fq Fr -> Fq

y_ :: MPoint c Curve25519 Fq Fr -> Fq

type MPoint = Point Montgomery

Montgomery points.

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 r q => Group (Element r q)

Defined in Group.Field

Methods

add :: Element r q -> Element r q -> Element r q

dbl :: Element r q -> Element r q

def :: Element r q -> Bool

gen :: Element r q

id :: Element r q

inv :: Element r q -> Element r q

mul' :: Element r q -> Integer -> Element r q

order :: Element r q -> Integer

rnd :: MonadRandom m => m (Element r q)

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)

Orphan instances

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

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 :: 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

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

Methods

pretty :: MAPoint e q r -> Doc

prettyList :: [MAPoint e q r] -> Doc

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

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)

Eq q => Eq (Point Montgomery Affine e q r)

Methods

(==) :: Point Montgomery Affine e q r -> Point Montgomery Affine e q r -> Bool

(/=) :: Point Montgomery Affine e q r -> Point Montgomery Affine e q r -> Bool

Read q => Read (Point Montgomery Affine e q r)

Methods

readsPrec :: Int -> ReadS (Point Montgomery Affine e q r)

readList :: ReadS [Point Montgomery Affine e q r]

readPrec :: ReadPrec (Point Montgomery Affine e q r)

readListPrec :: ReadPrec [Point Montgomery Affine e q r]

Show q => Show (Point Montgomery Affine e q r)

Methods

showsPrec :: Int -> Point Montgomery Affine e q r -> ShowS

show :: Point Montgomery Affine e q r -> String

showList :: [Point Montgomery Affine e q r] -> ShowS

Generic (Point Montgomery Affine e q r)

Associated Types

type Rep (Point Montgomery Affine e q r) :: Type -> Type

Methods

from :: Point Montgomery Affine e q r -> Rep (Point Montgomery Affine e q r) x

to :: Rep (Point Montgomery Affine e q r) x -> Point Montgomery Affine e q r

NFData q => NFData (Point Montgomery Affine e q r)

Methods

rnf :: Point Montgomery Affine e q r -> ()