An extensible library of elliptic curves used in cryptography research.
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 
is the point at infinity, and 
are K-rational coefficients that satisfy a non-zero discriminant condition. For cryptographic computational purposes, elliptic curves are represented in several different forms.
A (short) Weierstrass curve is an elliptic curve over 
for some prime 
, 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.
A (short Weierstrass) binary curve is an elliptic curve over 
for some positive 
, 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.
A Montgomery curve is an elliptic curve over for some prime , and is of the form
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.
A (twisted) Edwards curve is an elliptic curve over for some prime , 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.
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 FrArithmetic
-- 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 rSee Example.hs.
See DiffieHellman.hs.
See Weierstrass.
See Anomalous.
Import a curve implementation.
import qualified Data.Curve.Weierstrass.Anomalous as AnomalousThe 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 qThe following curves have already been implemented.
- SECT (NIST) curves
- Edwards curves
- Montgomery curves
- Anomalous
- ANSSIFRP256V1
- Barreto-Lynn-Scott (BLS) curves
- Barreto-Naehrig (BN) curves
- Brainpool curves
- SECP (NIST) curves
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.
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