Elliptic curve arithmetic: Difference between revisions

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=={{header|Haskell}}==

First, some ~~seful~~ imports:

First, some useful imports:

<lang haskell>import Data.Monoid

import Control.Monad (guard)

import Test.QuickCheck (quickCheck)</lang>

The datatype for a point on an elliptic curve ~~with exact zero~~:

The datatype for a point on an elliptic curve:

<lang haskell>import Data.Monoid

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We use QuickCheck to test general properties of points on arbitrary elliptic curve.

<lang haskell>-- for given a, b and x returns a point on the elliptic curve (~~it the~~ point exists)

<lang haskell>-- for given a, b and x returns a point on the positive branch of elliptic curve (if point exists)

~~ellipticY~~ a b Nothing = Just Zero

elliptic a b Nothing = Just Zero

~~ellipticY~~ a b (Just x) =

elliptic a b (Just x) =

do let y2 = x**3 + a*x + b

guard (y2 > 0)

return $ Elliptic x (sqrt y2)

addition a b x1 x2 =

let p = elliptic a b

s = p x1 <> p x2

in (s /= Nothing) ==> (s <> (inv <$> s) == Just Zero)

associativity a b x1 x2 x3 =

let p = ~~ellipticY~~ a b

let p = elliptic a b

in (p x1 <> p x2) <> p x3 == p x1 <> (p x2 <> p x3)

commutativity a b x1 x2 =

let p = ~~ellipticY~~ a b

let p = elliptic a b

in p x1 <> p x2 == p x2 <> p x1</lang>

<pre>λ> quickCheck ~~associativity~~

<pre>λ> quickCheck addition

+++ OK, passed 100 tests.

λ> quickCheck associativity

+++ OK, passed 100 tests.

λ> quickCheck commutativity