Data commit

Task/Elliptic-curve-arithmetic/Haskell/elliptic-curve-arithmetic-1.hs

@@ -0,0 +1,3 @@
+import Data.Monoid
+import Control.Monad (guard)
+import Test.QuickCheck (quickCheck)

Task/Elliptic-curve-arithmetic/Haskell/elliptic-curve-arithmetic-2.hs

@@ -0,0 +1,15 @@
+import Data.Monoid
+
+data Elliptic = Elliptic Double Double | Zero
+   deriving Show
+
+instance Eq Elliptic where
+  p == q = dist p q < 1e-14
+    where
+      dist Zero Zero = 0
+      dist Zero p = 1/0
+      dist p Zero = 1/0
+      dist (Elliptic x1 y1) (Elliptic x2 y2) = (x2-x1)^2 + (y2-y1)^2
+
+inv Zero = Zero
+inv (Elliptic x y) = Elliptic x (-y)

Task/Elliptic-curve-arithmetic/Haskell/elliptic-curve-arithmetic-3.hs

@@ -0,0 +1,13 @@
+instance Monoid Elliptic where
+  mempty = Zero
+
+  mappend Zero p = p
+  mappend p Zero = p
+  mappend p@(Elliptic x1 y1) q@(Elliptic x2 y2)
+    | p == inv q = Zero
+    | p == q     = mkElliptic $ 3*x1^2/(2*y1)
+    | otherwise  = mkElliptic $ (y2 - y1)/(x2 - x1)
+    where
+      mkElliptic l = let x = l^2 - x1 - x2
+                         y = l*(x1 - x) - y1
+                     in Elliptic x y

Task/Elliptic-curve-arithmetic/Haskell/elliptic-curve-arithmetic-4.hs

@@ -0,0 +1,2 @@
+ellipticX b y = Elliptic (qroot (y^2 - b)) y
+  where qroot x = signum x * abs x ** (1/3)

Task/Elliptic-curve-arithmetic/Haskell/elliptic-curve-arithmetic-5.hs

@@ -0,0 +1,2 @@
+mult :: Int -> Elliptic -> Elliptic
+mult n = mconcat . replicate n

Task/Elliptic-curve-arithmetic/Haskell/elliptic-curve-arithmetic-6.hs

@@ -0,0 +1,7 @@
+n `mult` p
+  | n == 0 = Zero
+  | n == 1 = p
+  | n == 2 = p <> p
+  | n < 0  = inv ((-n) `mult` p)
+  | even n = 2 `mult` ((n `div` 2) `mult` p)
+  | odd n  = p <> (n -1) `mult` p

Task/Elliptic-curve-arithmetic/Haskell/elliptic-curve-arithmetic-7.hs

@@ -0,0 +1,19 @@
+-- for given a, b and x returns a point on the positive branch of elliptic curve (if point exists)
+elliptic a b Nothing  = Just Zero
+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 = elliptic a b
+  in (p x1 <> p x2) <> p x3 == p x1 <> (p x2 <> p x3)
+
+commutativity a b x1 x2 =
+  let p = elliptic a b
+  in p x1 <> p x2 == p x2 <> p x1