    -- | Elliptic Curve Arithmetic.
    --
    -- /WARNING:/ These functions are vulnerable to timing attacks.
    module Crypto.PubKey.ECC.Prim
        ( scalarGenerate
        , pointAdd
        , pointNegate
        , pointDouble
        , pointBaseMul
        , pointMul
        , pointAddTwoMuls
        , isPointAtInfinity
        , isPointValid
        ) where

    import Data.Maybe
    import Crypto.Number.ModArithmetic
    import Crypto.Number.F2m
    import Crypto.Number.Generate (generateBetween)
    import Crypto.PubKey.ECC.Types
    import Crypto.Random

    -- | Generate a valid scalar for a specific Curve
    scalarGenerate :: MonadRandom randomly => Curve -> randomly PrivateNumber
    scalarGenerate curve = generateBetween 1 (n - 1)
      where
            n = ecc_n $ common_curve curve

    --TODO: Extract helper function for `fromMaybe PointO...`

    -- | Elliptic Curve point negation:
    -- @pointNegate c p@ returns point @q@ such that @pointAdd c p q == PointO@.
    pointNegate :: Curve -> Point -> Point
    pointNegate _            PointO     = PointO
    pointNegate (CurveFP c) (Point x y) = Point x (ecc_p c - y)
    pointNegate CurveF2m{}  (Point x y) = Point x (x `addF2m` y)

    -- | Elliptic Curve point addition.
    --
    -- /WARNING:/ Vulnerable to timing attacks.
    pointAdd :: Curve -> Point -> Point -> Point
    pointAdd _ PointO PointO = PointO
    pointAdd _ PointO q = q
    pointAdd _ p PointO = p
    pointAdd c p q
      | p == q = pointDouble c p
      | p == pointNegate c q = PointO
    pointAdd (CurveFP (CurvePrime pr _)) (Point xp yp) (Point xq yq)
        = fromMaybe PointO $ do
            s <- divmod (yp - yq) (xp - xq) pr
            let xr = (s ^ (2::Int) - xp - xq) `mod` pr
                yr = (s * (xp - xr) - yp) `mod` pr
            return $ Point xr yr
    pointAdd (CurveF2m (CurveBinary fx cc)) (Point xp yp) (Point xq yq)
        = fromMaybe PointO $ do
            s <- divF2m fx (yp `addF2m` yq) (xp `addF2m` xq)
            let xr = mulF2m fx s s `addF2m` s `addF2m` xp `addF2m` xq `addF2m` a
                yr = mulF2m fx s (xp `addF2m` xr) `addF2m` xr `addF2m` yp
            return $ Point xr yr
      where a = ecc_a cc

    -- | Elliptic Curve point doubling.
    --
    -- /WARNING:/ Vulnerable to timing attacks.
    --
    -- This perform the following calculation:
    -- > lambda = (3 * xp ^ 2 + a) / 2 yp
    -- > xr = lambda ^ 2 - 2 xp
    -- > yr = lambda (xp - xr) - yp
    --
    -- With binary curve:
    -- > xp == 0   => P = O
    -- > otherwise =>
    -- >    s = xp + (yp / xp)
    -- >    xr = s ^ 2 + s + a
    -- >    yr = xp ^ 2 + (s+1) * xr
    --
    pointDouble :: Curve -> Point -> Point
    pointDouble _ PointO = PointO
    pointDouble (CurveFP (CurvePrime pr cc)) (Point xp yp) = fromMaybe PointO $ do
        lambda <- divmod (3 * xp ^ (2::Int) + a) (2 * yp) pr
        let xr = (lambda ^ (2::Int) - 2 * xp) `mod` pr
            yr = (lambda * (xp - xr) - yp) `mod` pr
        return $ Point xr yr
      where a = ecc_a cc
    pointDouble (CurveF2m (CurveBinary fx cc)) (Point xp yp)
        | xp == 0   = PointO
        | otherwise = fromMaybe PointO $ do
            s <- return . addF2m xp =<< divF2m fx yp xp
            let xr = mulF2m fx s s `addF2m` s `addF2m` a
                yr = mulF2m fx xp xp `addF2m` mulF2m fx xr (s `addF2m` 1)
            return $ Point xr yr
      where a = ecc_a cc

    -- | Elliptic curve point multiplication using the base
    --
    -- /WARNING:/ Vulnerable to timing attacks.
    pointBaseMul :: Curve -> Integer -> Point
    pointBaseMul c n = pointMul c n (ecc_g $ common_curve c)

    -- | Elliptic curve point multiplication (double and add algorithm).
    --
    -- /WARNING:/ Vulnerable to timing attacks.
    pointMul :: Curve -> Integer -> Point -> Point
    pointMul _ _ PointO = PointO
    pointMul c n p
        | n <  0 = pointMul c (-n) (pointNegate c p)
        | n == 0 = PointO
        | n == 1 = p
        | odd n = pointAdd c p (pointMul c (n - 1) p)
        | otherwise = pointMul c (n `div` 2) (pointDouble c p)

    -- | Elliptic curve double-scalar multiplication (uses Shamir's trick).
    --
    -- > pointAddTwoMuls c n1 p1 n2 p2 == pointAdd c (pointMul c n1 p1)
    -- >                                             (pointMul c n2 p2)
    --
    -- /WARNING:/ Vulnerable to timing attacks.
    pointAddTwoMuls :: Curve -> Integer -> Point -> Integer -> Point -> Point
    pointAddTwoMuls _ _  PointO _  PointO = PointO
    pointAddTwoMuls c _  PointO n2 p2     = pointMul c n2 p2
    pointAddTwoMuls c n1 p1     _  PointO = pointMul c n1 p1
    pointAddTwoMuls c n1 p1     n2 p2
        | n1 < 0    = pointAddTwoMuls c (-n1) (pointNegate c p1) n2 p2
        | n2 < 0    = pointAddTwoMuls c n1 p1 (-n2) (pointNegate c p2)
        | otherwise = go (n1, n2)

      where
        p0 = pointAdd c p1 p2

        go (0,  0 ) = PointO
        go (k1, k2) =
            let q = pointDouble c $ go (k1 `div` 2, k2 `div` 2)
            in case (odd k1, odd k2) of
                (True  , True  ) -> pointAdd c p0 q
                (True  , False ) -> pointAdd c p1 q
                (False , True  ) -> pointAdd c p2 q
                (False , False ) -> q

    -- | Check if a point is the point at infinity.
    isPointAtInfinity :: Point -> Bool
    isPointAtInfinity PointO = True
    isPointAtInfinity _      = False

    -- | check if a point is on specific curve
    --
    -- This perform three checks:
    --
    -- * x is not out of range
    -- * y is not out of range
    -- * the equation @y^2 = x^3 + a*x + b (mod p)@ holds
    isPointValid :: Curve -> Point -> Bool
    isPointValid _                           PointO      = True
    isPointValid (CurveFP (CurvePrime p cc)) (Point x y) =
        isValid x && isValid y && (y ^ (2 :: Int)) `eqModP` (x ^ (3 :: Int) + a * x + b)
      where a  = ecc_a cc
            b  = ecc_b cc
            eqModP z1 z2 = (z1 `mod` p) == (z2 `mod` p)
            isValid e = e >= 0 && e < p
    isPointValid (CurveF2m (CurveBinary fx cc)) (Point x y) =
        and [ isValid x
            , isValid y
            , ((((x `add` a) `mul` x `add` y) `mul` x) `add` b `add` (squareF2m fx y)) == 0
            ]
      where a  = ecc_a cc
            b  = ecc_b cc
            add = addF2m
            mul = mulF2m fx
            isValid e = modF2m fx e == e

    -- | div and mod
    divmod :: Integer -> Integer -> Integer -> Maybe Integer
    divmod y x m = do
        i <- inverse (x `mod` m) m
        return $ y * i `mod` m