```{-# LANGUAGE PatternGuards,TypeOperators,FlexibleInstances,DatatypeContexts #-}
-- |
-- Module      :  Codec.Crypto.ECC.Base
-- Maintainer  :  Marcel FournÃ© (hecc@bitrot.dyndns.org)
--
-- ECC Base algorithms & point formats

module Codec.Crypto.ECC.Base (ECP(..),
EC(..),
modinv,
pmul,
ison,
binary,
--                            generateInteger,
EPa(..),
EPp(..),
EPj(..),
EPmj(..),
p256point,
p384point,
p521point,
ECPF2(..),
ECCNum(..),
ECurve(..),
ECSC(..),
modinvF2K,
pmulF2,
isonF2,
EPaF2(..),
EPpF2(..),
b283point,
k283point)
where

import Data.Bits
import Numeric
import Data.Char
import Data.List as L (length)
import Crypto.Types
-- import Crypto.Random
import Codec.Crypto.ECC.F2
import Codec.Crypto.ECC.StandardCurves
import qualified Data.Array.Repa as R

--
-- OLD Implementation, only for Integer
--

-- |extended euclidean algorithm, recursive variant
eeukl :: (Integral a ) => a -> a -> (a, a, a)
eeukl a 0 = (a,1,0)
eeukl a b = let (d,s,t) = eeukl b (a `mod` b)
in (d,t,s-(div a b)*t)

-- |computing the modular inverse of @a@ `mod` @m@
modinv :: (Integral a) => a -- ^the number to invert
-> a -- ^the modulus
-> a -- ^the inverted value
modinv a m = let (x,y,_) = eeukl a m
in if x == 1
then mod y m
else undefined

-- |class of all Elliptic Curves, has the form y^2=x^3+A*x+B mod P, the parameters being A, B and P
data EC = EC (Integer, Integer, Integer)
deriving (Eq)
instance Show EC where show (EC (a,b,p)) = "y^2=x^3+" ++ show a ++ "*x+" ++ show b ++ " mod " ++ show p

-- |class of all Elliptic Curve Points
class ECP a where
-- |function returning the appropriate INF in the specific ECP-Format, for generic higher-level-algorithms
inf :: a
-- |build point from one in affine coordinates
fromAffineCoords :: EPa -> a
-- |get bitlength
getBitLength :: a -> Int
-- |get contents of the curve
getCurve :: a -> EC
-- |generic getter, returning the affine x-value
getx :: a -> Integer
-- |generic getters, returning the affine y-value
gety :: a -> Integer
-- |add an elliptic point onto itself, base for padd a a
pdouble :: a -> a
padd :: a -> a -> a

-- |Elliptic Point Affine coordinates, two parameters x and y
data EPa = EPa (BitLength, EC, Integer, Integer)
| Infa
deriving (Eq)
instance Show EPa where show (EPa (a,b,c,d)) = show (a,b,c,d)
show Infa = "Null"
instance ECP EPa where
inf = Infa
fromAffineCoords = id
getBitLength (EPa (l,_,_,_)) = l
getBitLength (Infa) = undefined
getCurve (EPa (_,c,_,_)) = c
getCurve (Infa) = undefined
getx (EPa (_,_,x,_)) = x
getx Infa = undefined
gety (EPa (_,_,_,y)) = y
gety Infa = undefined
pdouble (EPa (l,c@(EC (alpha,_,p)),x1,y1)) =
let lambda = ((3*x1^(2::Int)+alpha)*(modinv (2*y1) p)) `mod` p
x3 = (lambda^(2::Int) - 2*x1) `mod` p
y3 = (lambda*(x1-x3)-y1) `mod` p
in EPa (l,c,x3,y3)
pdouble Infa = Infa
padd a@(EPa (l,c@(EC (_,_,p)),x1,y1)) b@(EPa (l',c',x2,y2))
| x1==x2,y1==(-y2) = Infa
| a==b = pdouble a
| otherwise =
let lambda = ((y2-y1)*(modinv (x2-x1) p)) `mod` p
x3 = (lambda^(2::Int) - x1 - x2) `mod` p
y3 = (lambda*(x1-x3)-y1) `mod` p
in if l==l' && c==c' then EPa (l,c,x3,y3)
else undefined

-- |Elliptic Point Projective coordinates, three parameters x, y and z, like affine (x/z,y/z)
data EPp = EPp (BitLength,EC,Integer, Integer, Integer)
| Infp
deriving (Eq)
instance Show EPp where show (EPp (a,b,c,d,e)) = show (a,b,c,d,e)
show Infp = "Null"
instance ECP EPp where
inf = Infp
fromAffineCoords (EPa (l,curve,a,b)) = EPp (l,curve,a,b,1)
fromAffineCoords Infa = Infp
getBitLength (EPp (l,_,_,_,_)) = l
getBitLength (Infp) = undefined
getCurve (EPp (_,c,_,_,_)) = c
getCurve (Infp) = undefined
getx (EPp (_,(EC (_,_,p)),x,_,z))= (x * (modinv z p)) `mod` p
getx Infp = undefined
gety (EPp (_,(EC (_,_,p)),_,y,z)) = (y * (modinv z p)) `mod` p
gety Infp = undefined
pdouble (EPp (l,curve@(EC (alpha,_,p)),x1,y1,z1)) =
let a = (alpha*z1^(2::Int)+3*x1^(2::Int)) `mod` p
b = (y1*z1) `mod` p
c = (x1*y1*b) `mod` p
d = (a^(2::Int)-8*c) `mod` p
x3 = (2*b*d) `mod` p
y3 = (a*(4*c-d)-8*y1^(2::Int)*b^(2::Int)) `mod` p
z3 = (8*b^(3::Int)) `mod` p
in EPp (l,curve,x3,y3,z3)
pdouble Infp = Infp
padd p1@(EPp (l,curve@(EC (_,_,p)),x1,y1,z1)) p2@(EPp (l',curve',x2,y2,z2))
| x1==x2,y1==(-y2) = Infp
| p1==p2 = pdouble p1
| otherwise =
let a = (y2*z1 - y1*z2) `mod` p
b = (x2*z1 - x1*z2) `mod` p
c = (a^(2::Int)*z1*z2 - b^(3::Int) - 2*b^(2::Int)*x1*z2) `mod` p
x3 = (b*c) `mod` p
y3 = (a*(b^(2::Int)*x1*z2-c)-b^(3::Int)*y1*z2) `mod` p
z3 = (b^(3::Int)*z1*z2) `mod` p
in if l==l' && curve==curve' then EPp (l,curve,x3,y3,z3)
else undefined

-- |Elliptic Point Jacobian coordinates, three parameter x, y and z, like affine (x/z^2,y/z^3)
data EPj = EPj (BitLength,EC,Integer, Integer, Integer)
| Infj
deriving (Eq)
instance Show EPj where show (EPj (a,b,c,d,e)) = show (a,b,c,d,e)
show Infj = "Null"
instance ECP EPj where
inf = Infj
fromAffineCoords (EPa (l,curve,a,b)) = EPj (l,curve,a,b,1)
fromAffineCoords Infa = Infj
getBitLength (EPj (l,_,_,_,_)) = l
getBitLength (Infj) = undefined
getCurve (EPj (_,c,_,_,_)) = c
getCurve (Infj) = undefined
getx (EPj (_,(EC (_,_,p)),x,_,z))= (x * (modinv (z^(2::Int)) p)) `mod` p
getx Infj = undefined
gety (EPj (_,(EC (_,_,p)),_,y,z)) = (y * (modinv (z^(3::Int)) p)) `mod` p
gety Infj = undefined
pdouble (EPj (l,c@(EC (alpha,_,p)),x1,y1,z1)) =
let a = 4*x1*y1^(2::Int) `mod` p
b = (3*x1^(2::Int) + alpha*z1^(4::Int)) `mod` p
x3 = (-2*a + b^(2::Int)) `mod` p
y3 = (-8*y1^(4::Int) + b*(a-x3)) `mod` p
z3 = 2*y1*z1 `mod` p
in EPj (l,c,x3,y3,z3)
pdouble Infj = Infj
padd p1@(EPj (l,curve@(EC (_,_,p)),x1,y1,z1)) p2@(EPj (l',curve',x2,y2,z2))
| x1==x2,y1==(-y2) = Infj
| p1==p2 = pdouble p1
| otherwise =
let a = (x1*z2^(2::Int)) `mod` p
b = (x2*z1^(2::Int)) `mod` p
c = (y1*z2^(3::Int)) `mod` p
d = (y2*z1^(3::Int)) `mod` p
e = (b - a) `mod` p
f = (d - c) `mod` p
x3 = (-e^(3::Int) - 2*a*e^(2::Int) + f^(2::Int)) `mod` p
y3 = (-c*e^(3::Int) + f*(a*e^(2::Int) - x3)) `mod` p
z3 = (z1*z2*e) `mod` p
in if l==l' && curve==curve' then EPj (l,curve,x3,y3,z3)
else undefined

-- |Elliptic Point Modified Jacobian coordinates, four parameters x,y,z and A*z^4 (A being the first curve-parameter), like affine coordinates (x/z^2,y/z^3)
data EPmj = EPmj (BitLength,EC,Integer, Integer, Integer, Integer)
| Infmj
deriving (Eq)
instance Show EPmj where show (EPmj (a,b,c,d,e,f)) = show (a,b,c,d,e,f)
show Infmj = "Null"
instance ECP EPmj where
inf = Infmj
fromAffineCoords (EPa (l,curve@(EC (alpha,_,_)),a,b)) = EPmj (l,curve,a,b,1,alpha)
fromAffineCoords Infa = Infmj
getBitLength (EPmj (l,_,_,_,_,_)) = l
getBitLength (Infmj) = undefined
getCurve (EPmj (_,c,_,_,_,_)) = c
getCurve (Infmj) = undefined
getx (EPmj (_,(EC (_,_,p)),x,_,z,_)) = (x * (modinv (z^(2::Int)) p)) `mod` p
getx Infmj = undefined
gety (EPmj (_,(EC (_,_,p)),_,y,z,_)) = (y * (modinv (z^(3::Int)) p)) `mod` p
gety Infmj = undefined
pdouble (EPmj (l,c@(EC (_,_,p)),x1,y1,z1,z1')) =
let s = 4*x1*y1^(2::Int) `mod` p
u = 8*y1^(4::Int) `mod` p
m = (3*x1^(2::Int) + z1') `mod` p
t = (-2*s + m^(2::Int)) `mod` p
x3 = t
y3 = (m*(s - t) - u) `mod` p
z3 = 2*y1*z1 `mod` p
z3' = 2*u*z1' `mod` p
in EPmj (l,c,x3,y3,z3,z3')
pdouble Infmj = Infmj
padd p1@(EPmj (l,curve@(EC (alpha,_,p)),x1,y1,z1,_)) p2@(EPmj (l',curve',x2,y2,z2,_))
| x1==x2,y1==(-y2) = Infmj
| p1==p2 = pdouble p1
| otherwise =
let u1 = (x1*z2^(2::Int)) `mod` p
u2 = (x2*z1^(2::Int)) `mod` p
s1 = (y1*z2^(3::Int)) `mod` p
s2 = (y2*z1^(3::Int)) `mod` p
h = (u2 - u1) `mod` p
r = (s2 - s1) `mod` p
x3 = (-h^(3::Int) - 2*u1*h^(2::Int) + r^(2::Int)) `mod` p
y3 = (-s1*h^(3::Int) + r*(u1*h^(2::Int) - x3)) `mod` p
z3 = (z1*z2*h) `mod` p
z3' = (alpha*z3^(4::Int)) `mod` p
in if l==l' && curve==curve' then EPmj (l,curve,x3,y3,z3,z3')
else undefined

-- |this is a generic handle for Point Multiplication. The implementation may change.
pmul :: (ECP a) => a -- ^the point to multiply
-> Integer -- ^times to multiply the point
-> a -- ^the result-point

-- |double and add for generic ECP
dnadd :: (ECP a) => a -> Integer -> a
let (EC (_,_,p)) = getCurve b
k = k' `mod` (p - 1)
ex a i
| i < 0 = a
| not (testBit k i) = ex (pdouble a) (i - 1)
| otherwise = ex (padd (pdouble a) b) (i - 1)
in ex inf (L.length (binary k) - 1)
-}

-- montgomery ladder, timing-attack-resistant (except for caches...)
montgladder :: (ECP a) => a -> Integer -> a
let (EC (_,_,p)) = getCurve b
k = k' `mod` (p - 1)
ex p1 p2 i
| i < 0 = p1
| not (testBit k i) = ex (pdouble p1) (padd p1 p2) (i - 1)
| otherwise = ex (padd p1 p2) (pdouble p2) (i - 1)
in ex b (pdouble b) ((L.length (binary k)) - 2)

-- binary representation of an integer
-- binary :: (Integral a) => a -> String
binary = flip (showIntAtBase 2 intToDigit) []

-- |generic verify, if generic ECP is on EC via getx and gety
ison :: (ECP a) => a -- ^ the elliptic curve point which we check
-> Bool -- ^is the point on the curve?
ison pt = let (EC (alpha,beta,p)) = getCurve pt
x = getx pt
y = gety pt
in (y^(2::Int)) `mod` p == (x^(3::Int)+alpha*x+beta) `mod` p
{-
-- | given a generator and a curve, generate a point randomly
genkey :: (ECP a) => a -- ^a generator (a point on the curve which multiplied gets to be every other point on the curve)
-> EC -- ^the curve
-> IO a -- ^the random point which will be the key
genkey a c@(EC (_,_,p)) = do
n <- evalRandIO \$ getRandomR (1,p)
return \$ pmul a n c
-}
{-
generateInteger :: (ECP a, CryptoRandomGen g) => a -> g -> Maybe (Integer, g)
generateInteger base g = let (EC (_,_,p)) = getCurve base
in case genInteger g (1,p-1) of
Left _ -> Nothing
Right (random1,g') -> Just (random1,g')
-}
-- helper-functions for getting basic points with less fuss
p521point :: (ECP a) => a
p521point = fromAffineCoords (EPa (stdc_l p521,(EC (stdc_a p521,stdc_b p521,stdc_p p521)), stdc_xp p521,stdc_xp p521))

p256point :: (ECP a) => a
p256point = fromAffineCoords (EPa (stdc_l p256,(EC (stdc_a p256,stdc_b p256,stdc_p p256)), stdc_xp p256,stdc_xp p256))

p384point :: (ECP a) => a
p384point = fromAffineCoords (EPa (stdc_l p384,(EC (stdc_a p384,stdc_b p384,stdc_p p384)), stdc_xp p384,stdc_xp p384))

--
-- NEW Implementation, for F(2^e)
--

-- platzhalter, falls aufteilen mehr bringt, ansonsten weiter montgladder
-- |computing the modular inverse of @a@ `emod` @m@
modinvF2K :: (ECPF2 a) => a -- ^the point to invert
-> a -- ^the inverted point
modinvF2K x = let d = getBitLengthF2 x
in pmulF2 x ((2^d)-2)

-- This class looks necessary, because repa has it's own Num-instance which is not what's wanted
class ECCNum a where
-- | abstract over (+)
eadd :: a -> a -> a
-- | abstract over (*)
emul :: a -> a -> a
-- | abstract over (^), used for small exponents
epow :: a -> Integer -> a
-- | abstract over mod
emod :: a -> a -> a

instance ECCNum (R.Array R.U R.DIM1 Bool) where
emul = f2eMul
epow = f2ePow
emod = f2eReduceBy

-- | All Elliptic Curves, binary
class ECurve a where
getA :: a -> R.Array R.U R.DIM1 Bool
getB :: a -> R.Array R.U R.DIM1 Bool
getP :: a -> R.Array R.U R.DIM1 Bool

-- |class of (non-hyper) Elliptic Curves, has the form y^2+x*y=x^3+A*x^2+B mod P, the parameters being A, B and P
data (ECCNum a) => ECSC a = ECSC (a, a, a)
deriving (Eq)
instance Show (ECSC (R.Array R.U R.DIM1 Bool)) where show (ECSC (a,b,p)) = "y^2+x*y=x^3+" ++ show ((f2eToInteger a)::Integer) ++ "*x^2+" ++ show ((f2eToInteger b)::Integer) ++ " mod " ++ show ((f2eToInteger p)::Integer)
instance ECurve (ECSC (R.Array R.U R.DIM1 Bool)) where
getA (ECSC (a,_,_)) = a
getB (ECSC (_,b,_)) = b
getP (ECSC (_,_,p)) = p

-- |class of all Elliptic Curve Points
class ECPF2 a where
-- |function returning the appropriate INF in the specific ECP-Format, for generic higher-level-algorithms
infF2 :: a
-- |build point from one in affine coordinates
fromAffineCoordsF2 :: EPaF2 -> a
-- |get bitlength
getBitLengthF2 :: a -> BitLength
-- |get contents of the curve
getCurveF2 :: a -> ECSC (R.Array R.U R.DIM1 Bool)
-- |generic getter, returning the affine x-value
getxF2 :: a -> R.Array R.U R.DIM1 Bool
-- |generic getters, returning the affine y-value
getyF2 :: a -> R.Array R.U R.DIM1 Bool
-- |add an elliptic point onto itself, base for padd a a
pdoubleF2 :: a -> a
paddF2 :: a -> a -> a

-- |Elliptic Point Affine coordinates, two parameters x and y
data EPaF2 = EPaF2 (BitLength, ECSC (R.Array R.U R.DIM1 Bool), R.Array R.U R.DIM1 Bool, R.Array R.U R.DIM1 Bool)
| InfaF2
deriving (Eq)
instance Show EPaF2 where show (EPaF2 (a,b,c,d)) = show (a,b,((f2eToInteger c)::Integer),((f2eToInteger d)::Integer))
show InfaF2 = "Null"
instance ECPF2 EPaF2 where
infF2 = InfaF2
fromAffineCoordsF2 = id
getBitLengthF2 (EPaF2 (l,_,_,_)) = l
getBitLengthF2 (InfaF2) = undefined
getCurveF2 (EPaF2 (_,c,_,_)) = c
getCurveF2 (InfaF2) = undefined
getxF2 (EPaF2 (_,_,x,_)) = x
getxF2 InfaF2 = undefined
getyF2 (EPaF2 (_,_,_,y)) = y
getyF2 InfaF2 = undefined
pdoubleF2 (EPaF2 (l,c@(ECSC (alpha,_,p)),x1,y1)) =
let lambda = (x1 `eadd` (y1 `emul` (modinvF2 x1 p)))
in EPaF2 (l,c,x3,y3)
pdoubleF2 InfaF2 = InfaF2
paddF2 a@(EPaF2 (l,c@(ECSC (alpha,_,p)),x1,y1)) b@(EPaF2 (l',c',x2,y2))
| ((f2eLen x1 == f2eLen x2) && (x1==x2)), (f2eLen y1 == f2eLen y2 && f2eLen x2 == f2eLen y2) && (y1==(x2 `eadd` y2)) = InfaF2
| (f2eLen x1 == f2eLen x2) && (f2eLen y1 == f2eLen y2) && a==b = pdoubleF2 a
| otherwise =
let lambda = ((y1 `eadd` y2) `emul` (modinvF2 (x1 `eadd` x2) p)) `emod` p
in if l==l' && c==c' then EPaF2 (l,c,x3,y3)
else undefined

-- |Elliptic Point Projective coordinates, three parameters x, y and z, like affine (x/z,y/z)
data EPpF2 = EPpF2 (BitLength, ECSC (R.Array R.U R.DIM1 Bool), R.Array R.U R.DIM1 Bool, R.Array R.U R.DIM1 Bool, R.Array R.U R.DIM1 Bool)
| InfpF2
deriving (Eq)
instance Show EPpF2 where show (EPpF2 (a,b,c,d,e)) = show (a,b,((f2eToInteger c)::Integer),((f2eToInteger d)::Integer),((f2eToInteger e)::Integer))
show InfpF2 = "Null"

instance ECPF2 EPpF2 where
infF2 = InfpF2
fromAffineCoordsF2 (EPaF2 (l,curve,a,b)) = EPpF2 (l,curve,a,b,f2eFromInteger 1)
fromAffineCoordsF2 InfaF2 = InfpF2
getBitLengthF2 (EPpF2 (l,_,_,_,_)) = l
getBitLengthF2 (InfpF2) = undefined
getCurveF2 (EPpF2 (_,c,_,_,_)) = c
getCurveF2 (InfpF2) = undefined
getxF2 (EPpF2 (_,(ECSC (_,_,p)),x,_,z))= (x `emul` (modinvF2 z p)) `emod` p
getxF2 InfpF2 = undefined
getyF2 (EPpF2 (_,(ECSC (_,_,p)),_,y,z)) = (y `emul` (modinvF2 z p)) `emod` p
getyF2 InfpF2 = undefined
pdoubleF2 (EPpF2 (l,curve@(ECSC (alpha,_,p)),x1,y1,z1)) =
let a = (x1 `epow` 2) `emod` p
b = (a `eadd` (y1 `emul` z1)) `emod` p
c = (x1 `emul` z1) `emod` p
d = (c `epow` 2) `emod` p
e = ((b `epow` 2) `eadd` (b `emul` c) `eadd` (alpha `emul` d)) `emod` p
x3 = (c `emul` e) `emod` p
y3 = (((b `eadd` c) `emul` e) `eadd` ((a `epow` 2) `emul` c)) `emod` p
z3 = (c `emul` d) `emod` p
in EPpF2 (l,curve,x3,y3,z3)
pdoubleF2 InfpF2 = InfpF2
paddF2 p1@(EPpF2 (l,curve@(ECSC (alpha,_,p)),x1,y1,z1)) p2@(EPpF2 (l',curve',x2,y2,z2))
| ((f2eLen x1 == f2eLen x2) && (x1==x2)),((f2eLen y1 == f2eLen y2 && f2eLen x2 == f2eLen y2) && y1==(x2 `eadd` y2)) = InfpF2
| (f2eLen x1 == f2eLen x2) && (f2eLen y1 == f2eLen y2) && p1==p2 = pdoubleF2 p1
| otherwise =
let a = ((y1 `emul` z2) `eadd` (z1 `emul` y2)) `emod` p
b = ((x1 `emul` z2)  `eadd`  (z1 `emul` x2)) `emod` p
c = (x1 `emul` z1) `emod` p
d = (c `epow` 2) `emod` p
e = ((((a `epow` 2) `eadd` (a `emul` b) `eadd` (alpha `emul` c)) `emul` d) `eadd` (b `emul` c)) `emod` p
x3 = (b `emul` e) `emod` p
y3 = (((c `emul` ((a `emul` x1) `eadd` (y1 `emul` b))) `emul` z2) `eadd` ((a `eadd` b) `emul` e)) `emod` p
z3 = ((b `epow` 3) `emul` d) `emod` p
in if l==l' && curve==curve' then EPpF2 (l,curve,x3,y3,z3)
else undefined

-- |this is a generic handle for Point Multiplication. The implementation may change.
pmulF2 :: (ECPF2 a) => a -- ^the point to multiply
-> Integer -- ^times to multiply the point
-> (ECPF2 a) => a -- ^the result-point

-- montgomery ladder, timing-attack-resistant (except for caches...)
montgladderF2 :: (ECPF2 a) => a -> Integer -> a
let (ECSC (_,_,p)) = getCurveF2 b
k = k' `mod` ((f2eToInteger p) - 1)
ex p1 p2 i
| i < 0 = p1
| not (testBit k i) = ex (pdoubleF2 p1) (paddF2 p1 p2) (i - 1)
| otherwise = ex (paddF2 p1 p2) (pdoubleF2 p2) (i - 1)
in ex b (pdoubleF2 b) ((L.length (binary k)) - 2)

-- |generic verify, if generic ECP is on EC via getx and gety
isonF2 :: (ECPF2 a, Eq a) => a -- ^ the elliptic curve point which we check
-> Bool -- ^is the point on the curve?
isonF2 pt = let (ECSC (alpha,beta,p)) = getCurveF2 pt
x = getxF2 pt
y = getyF2 pt
in ((y `epow` 2) `eadd` (x `emul` y)) `emod` p == ((x `epow` 3) `eadd` (alpha `emul` (x `epow` 2)) `eadd` beta) `emod` p

b283point :: (ECPF2 a) => a
b283point = fromAffineCoordsF2 (EPaF2 (stdcF_l b283,(ECSC (stdcF_a b283,stdcF_b b283,stdcF_p b283)), stdcF_xp b283,stdcF_yp b283))

k283point :: (ECPF2 a) => a
k283point = fromAffineCoordsF2 (EPaF2 (stdcF_l k283,(ECSC (stdcF_a k283,stdcF_b k283,stdcF_p k283)), stdcF_xp k283,stdcF_yp k283))
```