elliptic-curve

A polymorphic interface for elliptic curve operations

Monthly Downloads: 3

Programming language: Haskell

License: MIT License

Tags: Cryptography    

Latest version: v0.3.1

README

Elliptic Curve

An extensible library of elliptic curves used in cryptography research.

Curve representations

An [elliptic curve](src/Data/Curve.hs) 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.

Weierstrass curves

A (short) [Weierstrass curve](src/Data/Curve/Weierstrass.hs) 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.

Binary curves

A (short Weierstrass) [binary curve](src/Data/Curve/Binary.hs) 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.

Montgomery curves

A [Montgomery curve](src/Data/Curve/Montgomery.hs) 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.

Edwards curves

A (twisted) [Edwards curve](src/Data/Curve/Edwards.hs) 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.

Curve usage

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

• Binary
• Edwards
• Montgomery
• Weierstrass

Coordinates

• Affine
• Jacobian
• Projective

These are then specialised down into type classes for the different forms.

• BCurve
• ECurve
• MCurve
• WCurve

And then by coordinate system.

• BACurve
• BPCurve
• EACurve
• EPCurve
• MACurve
• WACurve
• WJCurve
• WPCurve

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:

```haskell ignore 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 Fr

**Arithmetic**

```haskell ignore
-- 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

```haskell ignore -- 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 r

### Point Arithmetic

See [**Example.hs**](examples/Example.hs).

### Elliptic Curve Diffie-Hellman (ECDH)

See [**DiffieHellman.hs**](examples/DiffieHellman.hs).

### Representing a new curve using the curve class

See [**Weierstrass**](src/Data/Curve/Weierstrass.hs).

### Implementing a new curve using a curve representation

See [**Anomalous**](src/Data/Curve/Weierstrass/Anomalous.hs).

### Using an implemented curve

Import a curve implementation.

```haskell
import qualified Data.Curve.Weierstrass.Anomalous as Anomalous

The 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 q

Curve implementations

The following curves have already been implemented.

Binary curves

• SECT (NIST) curves
  • [SECT113R1](src/Data/Curve/Binary/SECT113R1.hs)
  • [SECT113R2](src/Data/Curve/Binary/SECT113R2.hs)
  • [SECT131R1](src/Data/Curve/Binary/SECT131R1.hs)
  • [SECT131R2](src/Data/Curve/Binary/SECT131R2.hs)
  • [SECT163K1](src/Data/Curve/Binary/SECT163K1.hs)
  • [SECT163R1](src/Data/Curve/Binary/SECT163R1.hs)
  • [SECT163R2](src/Data/Curve/Binary/SECT163R2.hs)
  • [SECT193R1](src/Data/Curve/Binary/SECT193R1.hs)
  • [SECT193R2](src/Data/Curve/Binary/SECT193R2.hs)
  • [SECT233K1](src/Data/Curve/Binary/SECT233K1.hs)
  • [SECT233R1](src/Data/Curve/Binary/SECT233R1.hs)
  • [SECT239K1](src/Data/Curve/Binary/SECT239K1.hs)
  • [SECT283K1](src/Data/Curve/Binary/SECT283K1.hs)
  • [SECT283R1](src/Data/Curve/Binary/SECT283R1.hs)
  • [SECT409K1](src/Data/Curve/Binary/SECT409K1.hs)
  • [SECT409R1](src/Data/Curve/Binary/SECT409R1.hs)
  • [SECT571K1](src/Data/Curve/Binary/SECT571K1.hs)
  • [SECT571R1](src/Data/Curve/Binary/SECT571R1.hs)

Edwards curves

• Edwards curves
  • [Curve1174](src/Data/Curve/Edwards/Curve1174.hs)
  • [Curve41417](src/Data/Curve/Edwards/Curve41417.hs)
  • [E-222](src/Data/Curve/Edwards/E222.hs)
  • [E-382](src/Data/Curve/Edwards/E382.hs)
  • [E-521](src/Data/Curve/Edwards/E521.hs)
  • [Ed25519 (Curve25519)](src/Data/Curve/Edwards/Ed25519.hs)
  • [Ed3363 (HighFive)](src/Data/Curve/Edwards/Ed3363.hs)
  • [Ed448 (Goldilocks)](src/Data/Curve/Edwards/Ed448.hs)
  • [JubJub](src/Data/Curve/Edwards/JubJub.hs)

Montgomery curves

• Montgomery curves
  • [Curve25519 (Ed25519)](src/Data/Curve/Montgomery/Curve25519.hs)
  • [Curve383187](src/Data/Curve/Montgomery/Curve383187.hs)
  • [Curve448 (Goldilocks)](src/Data/Curve/Montgomery/Curve448.hs)
  • [M221](src/Data/Curve/Montgomery/M221.hs)
  • [M383](src/Data/Curve/Montgomery/M383.hs)
  • [M511](src/Data/Curve/Montgomery/M511.hs)

Weierstrass curves

• [Anomalous](src/Data/Curve/Weierstrass/Anomalous.hs)
• [ANSSIFRP256V1](src/Data/Curve/Weierstrass/ANSSIFRP256V1.hs)
• Barreto-Lynn-Scott (BLS) curves
  • [BLS12381](src/Data/Curve/Weierstrass/BLS12381.hs)
  • [BLS12381T](src/Data/Curve/Weierstrass/BLS12381T.hs)
  • [BLS48581](src/Data/Curve/Weierstrass/BLS48581.hs)
  • [BLS48581T](src/Data/Curve/Weierstrass/BLS48581T.hs)
• Barreto-Naehrig (BN) curves
  • [BN224 (Fp224BN)](src/Data/Curve/Weierstrass/BN224.hs)
  • [BN254 (CurveSNARK)](src/Data/Curve/Weierstrass/BN254.hs)
  • [BN254T (CurveSNARKn2)](src/Data/Curve/Weierstrass/BN254T.hs)
  • [BN254A (Fp254BNa)](src/Data/Curve/Weierstrass/BN254A.hs)
  • [BN254AT (Fp254n2BNa)](src/Data/Curve/Weierstrass/BN254AT.hs)
  • [BN254B (Fp254BNb)](src/Data/Curve/Weierstrass/BN254B.hs)
  • [BN254BT (Fp254n2BNb)](src/Data/Curve/Weierstrass/BN254BT.hs)
  • [BN254C (Fp254BNc)](src/Data/Curve/Weierstrass/BN254C.hs)
  • [BN254CT (Fp254n2BNc)](src/Data/Curve/Weierstrass/BN254CT.hs)
  • [BN254D (Fp254BNd)](src/Data/Curve/Weierstrass/BN254D.hs)
  • [BN254DT (Fp254n2BNd)](src/Data/Curve/Weierstrass/BN254DT.hs)
  • [BN256 (Fp256BN)](src/Data/Curve/Weierstrass/BN256.hs)
  • [BN384 (Fp384BN)](src/Data/Curve/Weierstrass/BN384.hs)
  • [BN462 (Fp462BN)](src/Data/Curve/Weierstrass/BN462.hs)
  • [BN462T (Fp462n2BN)](src/Data/Curve/Weierstrass/BN462T.hs)
  • [BN512 (Fp512BN)](src/Data/Curve/Weierstrass/BN512.hs)
• Brainpool curves
  • [BrainpoolP160R1](src/Data/Curve/Weierstrass/BrainpoolP160R1.hs)
  • [BrainpoolP160T1](src/Data/Curve/Weierstrass/BrainpoolP160T1.hs)
  • [BrainpoolP192R1](src/Data/Curve/Weierstrass/BrainpoolP192R1.hs)
  • [BrainpoolP192T1](src/Data/Curve/Weierstrass/BrainpoolP192T1.hs)
  • [BrainpoolP224R1](src/Data/Curve/Weierstrass/BrainpoolP224R1.hs)
  • [BrainpoolP224T1](src/Data/Curve/Weierstrass/BrainpoolP224T1.hs)
  • [BrainpoolP256R1](src/Data/Curve/Weierstrass/BrainpoolP256R1.hs)
  • [BrainpoolP256T1](src/Data/Curve/Weierstrass/BrainpoolP256T1.hs)
  • [BrainpoolP320R1](src/Data/Curve/Weierstrass/BrainpoolP320R1.hs)
  • [BrainpoolP320T1](src/Data/Curve/Weierstrass/BrainpoolP320T1.hs)
  • [BrainpoolP384R1](src/Data/Curve/Weierstrass/BrainpoolP384R1.hs)
  • [BrainpoolP384T1](src/Data/Curve/Weierstrass/BrainpoolP384T1.hs)
  • [BrainpoolP512R1](src/Data/Curve/Weierstrass/BrainpoolP512R1.hs)
  • [BrainpoolP512T1](src/Data/Curve/Weierstrass/BrainpoolP512T1.hs)
• SECP (NIST) curves
  • [SECP112R1](src/Data/Curve/Weierstrass/SECP112R1.hs)
  • [SECP112R2](src/Data/Curve/Weierstrass/SECP112R2.hs)
  • [SECP128R1](src/Data/Curve/Weierstrass/SECP128R1.hs)
  • [SECP128R2](src/Data/Curve/Weierstrass/SECP128R2.hs)
  • [SECP160K1](src/Data/Curve/Weierstrass/SECP160K1.hs)
  • [SECP160R1](src/Data/Curve/Weierstrass/SECP160R1.hs)
  • [SECP160R2](src/Data/Curve/Weierstrass/SECP160R2.hs)
  • [SECP192K1](src/Data/Curve/Weierstrass/SECP192K1.hs)
  • [SECP192R1](src/Data/Curve/Weierstrass/SECP192R1.hs)
  • [SECP224K1](src/Data/Curve/Weierstrass/SECP224K1.hs)
  • [SECP224R1](src/Data/Curve/Weierstrass/SECP224R1.hs)
  • [SECP256K1](src/Data/Curve/Weierstrass/SECP256K1.hs)
  • [SECP256R1](src/Data/Curve/Weierstrass/SECP256R1.hs)
  • [SECP384R1](src/Data/Curve/Weierstrass/SECP384R1.hs)
  • [SECP521R1](src/Data/Curve/Weierstrass/SECP521R1.hs)

Disclaimer

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.

License

Copyright (c) 2019-2020 Adjoint Inc.

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