| Safe Haskell | Safe-Inferred |
|---|---|
| Language | Haskell2010 |
ZkFold.Base.Algebra.EllipticCurve.Class
Synopsis
- class (Field (BaseFieldOf point), Eq (BaseFieldOf point), Planar (BaseFieldOf point) point, AdditiveGroup point) => EllipticCurve point where
- type CurveOf point :: Symbol
- type BaseFieldOf point :: Type
- isOnCurve :: point -> BooleanOf (BaseFieldOf point)
- class (AdditiveGroup g, FiniteField (ScalarFieldOf g), Scale (ScalarFieldOf g) g) => CyclicGroup g where
- type ScalarFieldOf g :: Type
- pointGen :: g
- type CycleOfCurves g1 g2 = (EllipticCurve g1, EllipticCurve g2, CyclicGroup g1, CyclicGroup g2, ScalarFieldOf g1 ~ BaseFieldOf g2, BaseFieldOf g1 ~ ScalarFieldOf g2)
- class Field field => WeierstrassCurve (curve :: Symbol) field where
- weierstrassB :: field
- class Field field => TwistedEdwardsCurve (curve :: Symbol) field where
- twistedEdwardsA :: field
- twistedEdwardsD :: field
- class Eq (BaseFieldOf point) => Compressible point where
- type Compressed point :: Type
- pointCompressed :: BaseFieldOf point -> BooleanOf (BaseFieldOf point) -> Compressed point
- compress :: point -> Compressed point
- decompress :: Compressed point -> point
- class (CyclicGroup g1, CyclicGroup g2, ScalarFieldOf g1 ~ ScalarFieldOf g2, MultiplicativeGroup gt, Exponent gt (ScalarFieldOf g1)) => Pairing g1 g2 gt | g1 g2 -> gt where
- pairing :: g1 -> g2 -> gt
- class Planar field point | point -> field where
- pointXY :: field -> field -> point
- class HasPointInf point where
- pointInf :: point
- newtype Weierstrass curve point = Weierstrass {
- pointWeierstrass :: point
- newtype TwistedEdwards curve point = TwistedEdwards {
- pointTwistedEdwards :: point
- data Point field = Point {}
- data CompressedPoint field = CompressedPoint {}
- data AffinePoint field = AffinePoint {}
curve classes
class (Field (BaseFieldOf point), Eq (BaseFieldOf point), Planar (BaseFieldOf point) point, AdditiveGroup point) => EllipticCurve point where Source #
Elliptic curves are plane algebraic curves that form AdditiveGroups.
Elliptic curves always have genus 1 and are birationally equivalent
to a projective curve of degree 3. As such, elliptic curves are
the simplest curves after conic sections, curves of degree 2,
and lines, curves of degree 1. Bézout's theorem implies
that a line in general position will intersect with an
elliptic curve at 3 points counting multiplicity;
point0, point1 and point2.
The geometric group law of the elliptic curve is:
point0 + point1 + point2 = zero
Methods
isOnCurve :: point -> BooleanOf (BaseFieldOf point) Source #
isOnCurve validates an equation for a plane algebraic curve
which has degree 3 up to some birational equivalence.
Instances
| (TwistedEdwardsCurve curve field, Field field, Eq field) => EllipticCurve (TwistedEdwards curve (AffinePoint field)) Source # | |
Defined in ZkFold.Base.Algebra.EllipticCurve.Class Associated Types type CurveOf (TwistedEdwards curve (AffinePoint field)) :: Symbol Source # type BaseFieldOf (TwistedEdwards curve (AffinePoint field)) Source # Methods isOnCurve :: TwistedEdwards curve (AffinePoint field) -> BooleanOf (BaseFieldOf (TwistedEdwards curve (AffinePoint field))) Source # | |
| (WeierstrassCurve curve field, Conditional (BooleanOf field) (BooleanOf field), Conditional (BooleanOf field) field, Eq field, Field field) => EllipticCurve (Weierstrass curve (Point field)) Source # | |
Defined in ZkFold.Base.Algebra.EllipticCurve.Class Associated Types type CurveOf (Weierstrass curve (Point field)) :: Symbol Source # type BaseFieldOf (Weierstrass curve (Point field)) Source # Methods isOnCurve :: Weierstrass curve (Point field) -> BooleanOf (BaseFieldOf (Weierstrass curve (Point field))) Source # | |
class (AdditiveGroup g, FiniteField (ScalarFieldOf g), Scale (ScalarFieldOf g) g) => CyclicGroup g where Source #
Both the ECDSA and ECDH algorithms make use of
the elliptic curve discrete logarithm problem, ECDLP.
There may be a discrete "exponential" function
from a PrimeField of scalars
into the AdditiveGroup of points on an elliptic curve.
It's given naturally by scaling a point of prime order,
if there is one on the curve.
scale order pointGen = zero
>>>let discreteExp scalar = scale scalar pointGen
Then the inverse of discreteExp is hard to compute.
Associated Types
type ScalarFieldOf g :: Type Source #
Methods
generator of a cyclic subgroup
scale (order @(ScalarFieldOf g)) pointGen = zero
Instances
type CycleOfCurves g1 g2 = (EllipticCurve g1, EllipticCurve g2, CyclicGroup g1, CyclicGroup g2, ScalarFieldOf g1 ~ BaseFieldOf g2, BaseFieldOf g1 ~ ScalarFieldOf g2) Source #
A cycle of two curves elliptic curves over finite fields such that the number of points on one curve is equal to the size of the field of definition of the next, in a cyclic way.
class Field field => WeierstrassCurve (curve :: Symbol) field where Source #
The standard form of an elliptic curve is the Weierstrass equation:
y^2 = x^3 + a*x + b
- Weierstrass curves have x-axis symmetry.
- The characteristic of the field must not be
2or3. - The curve must have nonzero discriminant
Δ = -16 * (4*a^3 + 27*b^3). - When
a = 0some computations can be simplified so all the public Weierstrass curves havea = zeroand nonzeroband we do too.
Methods
weierstrassB :: field Source #
Instances
class Field field => TwistedEdwardsCurve (curve :: Symbol) field where Source #
A twisted Edwards curve is defined by the equation:
a*x^2 + y^2 = 1 + d*x^2*y^2
- Twisted Edwards curves have y-axis symmetry.
- The characteristic of the field must not be
2. aanddmust be nonzero.
Instances
| Field field => TwistedEdwardsCurve "ed25519" field Source # | |
Defined in ZkFold.Base.Algebra.EllipticCurve.Ed25519 | |
class Eq (BaseFieldOf point) => Compressible point where Source #
Associated Types
type Compressed point :: Type Source #
Methods
pointCompressed :: BaseFieldOf point -> BooleanOf (BaseFieldOf point) -> Compressed point Source #
compress :: point -> Compressed point Source #
decompress :: Compressed point -> point Source #
Instances
| Compressible BLS12_381_G1_Point Source # | |
Defined in ZkFold.Base.Algebra.EllipticCurve.BLS12_381 Associated Types | |
| Compressible BLS12_381_G2_Point Source # | |
Defined in ZkFold.Base.Algebra.EllipticCurve.BLS12_381 Associated Types | |
class (CyclicGroup g1, CyclicGroup g2, ScalarFieldOf g1 ~ ScalarFieldOf g2, MultiplicativeGroup gt, Exponent gt (ScalarFieldOf g1)) => Pairing g1 g2 gt | g1 g2 -> gt where Source #
Instances
| Pairing BLS12_381_G1_Point BLS12_381_G2_Point BLS12_381_GT Source # | |
Defined in ZkFold.Base.Algebra.EllipticCurve.BLS12_381 Methods pairing :: BLS12_381_G1_Point -> BLS12_381_G2_Point -> BLS12_381_GT Source # | |
| Pairing BN254_G1_Point BN254_G2_Point BN254_GT Source # | |
Defined in ZkFold.Base.Algebra.EllipticCurve.BN254 Methods pairing :: BN254_G1_Point -> BN254_G2_Point -> BN254_GT Source # | |
point classes
class Planar field point | point -> field where Source #
A class for smart constructor method
pointXY for constructing points from an x and y coordinate.
Instances
| Planar field (AffinePoint field) Source # | |
Defined in ZkFold.Base.Algebra.EllipticCurve.Class Methods pointXY :: field -> field -> AffinePoint field Source # | |
| Eq field => Planar field (Point field) Source # | |
Defined in ZkFold.Base.Algebra.EllipticCurve.Class | |
| Planar field point => Planar field (TwistedEdwards curve point) Source # | |
Defined in ZkFold.Base.Algebra.EllipticCurve.Class Methods pointXY :: field -> field -> TwistedEdwards curve point Source # | |
| Planar field point => Planar field (Weierstrass curve point) Source # | |
Defined in ZkFold.Base.Algebra.EllipticCurve.Class Methods pointXY :: field -> field -> Weierstrass curve point Source # | |
class HasPointInf point where Source #
A class for smart constructor method
pointInf for constructing the point at infinity.
Instances
| (BoolType (BooleanOf field), AdditiveMonoid field) => HasPointInf (CompressedPoint field) Source # | |
Defined in ZkFold.Base.Algebra.EllipticCurve.Class Methods pointInf :: CompressedPoint field Source # | |
| (Semiring field, Eq field) => HasPointInf (Point field) Source # | |
Defined in ZkFold.Base.Algebra.EllipticCurve.Class | |
| HasPointInf point => HasPointInf (TwistedEdwards curve point) Source # | |
Defined in ZkFold.Base.Algebra.EllipticCurve.Class Methods pointInf :: TwistedEdwards curve point Source # | |
| HasPointInf point => HasPointInf (Weierstrass curve point) Source # | |
Defined in ZkFold.Base.Algebra.EllipticCurve.Class Methods pointInf :: Weierstrass curve point Source # | |
point types
newtype Weierstrass curve point Source #
Weierstrass tags a ProjectivePlanar point, over a Field field,
with a phantom WeierstrassCurve curve.
Constructors
| Weierstrass | |
Fields
| |
Instances
newtype TwistedEdwards curve point Source #
TwistedEdwards tags a Planar point, over a Field field,
with a phantom TwistedEdwardsCurve curve.
Constructors
| TwistedEdwards | |
Fields
| |
Instances
A type of points in the projective plane.
Instances
data CompressedPoint field Source #
Instances
data AffinePoint field Source #
Constructors
| AffinePoint | |