-- |
-- Module      : Crypto.PubKey.DSA
-- License     : BSD-style
-- Maintainer  : Vincent Hanquez <vincent@snarc.org>
-- Stability   : experimental
-- Portability : Good
--
-- An implementation of the Digital Signature Algorithm (DSA)
{-# LANGUAGE DeriveDataTypeable #-}
module Crypto.PubKey.DSA
    ( Params(..)
    , Signature(..)
    , PublicKey(..)
    , PrivateKey(..)
    , PublicNumber
    , PrivateNumber
    -- * Generation
    , generatePrivate
    , calculatePublic
    -- * Signature primitive
    , sign
    , signWith
    -- * Verification primitive
    , verify
    -- * Key pair
    , KeyPair(..)
    , toPublicKey
    , toPrivateKey
    ) where


import Data.Data
import Data.Maybe

import Crypto.Number.ModArithmetic (expFast, expSafe, inverse)
import Crypto.Number.Generate
import Crypto.Internal.ByteArray (ByteArrayAccess)
import Crypto.Internal.Imports
import Crypto.Hash
import Crypto.PubKey.Internal (dsaTruncHash)
import Crypto.Random.Types

-- | DSA Public Number, usually embedded in DSA Public Key
type PublicNumber = Integer

-- | DSA Private Number, usually embedded in DSA Private Key
type PrivateNumber = Integer

-- | Represent DSA parameters namely P, G, and Q.
data Params = Params
    { Params -> Integer
params_p :: Integer -- ^ DSA p
    , Params -> Integer
params_g :: Integer -- ^ DSA g
    , Params -> Integer
params_q :: Integer -- ^ DSA q
    } deriving (Int -> Params -> ShowS
[Params] -> ShowS
Params -> String
forall a.
(Int -> a -> ShowS) -> (a -> String) -> ([a] -> ShowS) -> Show a
showList :: [Params] -> ShowS
$cshowList :: [Params] -> ShowS
show :: Params -> String
$cshow :: Params -> String
showsPrec :: Int -> Params -> ShowS
$cshowsPrec :: Int -> Params -> ShowS
Show,ReadPrec [Params]
ReadPrec Params
Int -> ReadS Params
ReadS [Params]
forall a.
(Int -> ReadS a)
-> ReadS [a] -> ReadPrec a -> ReadPrec [a] -> Read a
readListPrec :: ReadPrec [Params]
$creadListPrec :: ReadPrec [Params]
readPrec :: ReadPrec Params
$creadPrec :: ReadPrec Params
readList :: ReadS [Params]
$creadList :: ReadS [Params]
readsPrec :: Int -> ReadS Params
$creadsPrec :: Int -> ReadS Params
Read,Params -> Params -> Bool
forall a. (a -> a -> Bool) -> (a -> a -> Bool) -> Eq a
/= :: Params -> Params -> Bool
$c/= :: Params -> Params -> Bool
== :: Params -> Params -> Bool
$c== :: Params -> Params -> Bool
Eq,Typeable Params
Params -> DataType
Params -> Constr
(forall b. Data b => b -> b) -> Params -> Params
forall a.
Typeable a
-> (forall (c :: * -> *).
    (forall d b. Data d => c (d -> b) -> d -> c b)
    -> (forall g. g -> c g) -> a -> c a)
-> (forall (c :: * -> *).
    (forall b r. Data b => c (b -> r) -> c r)
    -> (forall r. r -> c r) -> Constr -> c a)
-> (a -> Constr)
-> (a -> DataType)
-> (forall (t :: * -> *) (c :: * -> *).
    Typeable t =>
    (forall d. Data d => c (t d)) -> Maybe (c a))
-> (forall (t :: * -> * -> *) (c :: * -> *).
    Typeable t =>
    (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c a))
-> ((forall b. Data b => b -> b) -> a -> a)
-> (forall r r'.
    (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> a -> r)
-> (forall r r'.
    (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> a -> r)
-> (forall u. (forall d. Data d => d -> u) -> a -> [u])
-> (forall u. Int -> (forall d. Data d => d -> u) -> a -> u)
-> (forall (m :: * -> *).
    Monad m =>
    (forall d. Data d => d -> m d) -> a -> m a)
-> (forall (m :: * -> *).
    MonadPlus m =>
    (forall d. Data d => d -> m d) -> a -> m a)
-> (forall (m :: * -> *).
    MonadPlus m =>
    (forall d. Data d => d -> m d) -> a -> m a)
-> Data a
forall u. Int -> (forall d. Data d => d -> u) -> Params -> u
forall u. (forall d. Data d => d -> u) -> Params -> [u]
forall r r'.
(r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Params -> r
forall r r'.
(r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Params -> r
forall (m :: * -> *).
Monad m =>
(forall d. Data d => d -> m d) -> Params -> m Params
forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> Params -> m Params
forall (c :: * -> *).
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c Params
forall (c :: * -> *).
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> Params -> c Params
forall (t :: * -> *) (c :: * -> *).
Typeable t =>
(forall d. Data d => c (t d)) -> Maybe (c Params)
forall (t :: * -> * -> *) (c :: * -> *).
Typeable t =>
(forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c Params)
gmapMo :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> Params -> m Params
$cgmapMo :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> Params -> m Params
gmapMp :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> Params -> m Params
$cgmapMp :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> Params -> m Params
gmapM :: forall (m :: * -> *).
Monad m =>
(forall d. Data d => d -> m d) -> Params -> m Params
$cgmapM :: forall (m :: * -> *).
Monad m =>
(forall d. Data d => d -> m d) -> Params -> m Params
gmapQi :: forall u. Int -> (forall d. Data d => d -> u) -> Params -> u
$cgmapQi :: forall u. Int -> (forall d. Data d => d -> u) -> Params -> u
gmapQ :: forall u. (forall d. Data d => d -> u) -> Params -> [u]
$cgmapQ :: forall u. (forall d. Data d => d -> u) -> Params -> [u]
gmapQr :: forall r r'.
(r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Params -> r
$cgmapQr :: forall r r'.
(r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Params -> r
gmapQl :: forall r r'.
(r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Params -> r
$cgmapQl :: forall r r'.
(r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Params -> r
gmapT :: (forall b. Data b => b -> b) -> Params -> Params
$cgmapT :: (forall b. Data b => b -> b) -> Params -> Params
dataCast2 :: forall (t :: * -> * -> *) (c :: * -> *).
Typeable t =>
(forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c Params)
$cdataCast2 :: forall (t :: * -> * -> *) (c :: * -> *).
Typeable t =>
(forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c Params)
dataCast1 :: forall (t :: * -> *) (c :: * -> *).
Typeable t =>
(forall d. Data d => c (t d)) -> Maybe (c Params)
$cdataCast1 :: forall (t :: * -> *) (c :: * -> *).
Typeable t =>
(forall d. Data d => c (t d)) -> Maybe (c Params)
dataTypeOf :: Params -> DataType
$cdataTypeOf :: Params -> DataType
toConstr :: Params -> Constr
$ctoConstr :: Params -> Constr
gunfold :: forall (c :: * -> *).
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c Params
$cgunfold :: forall (c :: * -> *).
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c Params
gfoldl :: forall (c :: * -> *).
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> Params -> c Params
$cgfoldl :: forall (c :: * -> *).
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> Params -> c Params
Data)

instance NFData Params where
    rnf :: Params -> ()
rnf (Params Integer
p Integer
g Integer
q) = Integer
p seq :: forall a b. a -> b -> b
`seq` Integer
g seq :: forall a b. a -> b -> b
`seq` Integer
q seq :: forall a b. a -> b -> b
`seq` ()

-- | Represent a DSA signature namely R and S.
data Signature = Signature
    { Signature -> Integer
sign_r :: Integer -- ^ DSA r
    , Signature -> Integer
sign_s :: Integer -- ^ DSA s
    } deriving (Int -> Signature -> ShowS
[Signature] -> ShowS
Signature -> String
forall a.
(Int -> a -> ShowS) -> (a -> String) -> ([a] -> ShowS) -> Show a
showList :: [Signature] -> ShowS
$cshowList :: [Signature] -> ShowS
show :: Signature -> String
$cshow :: Signature -> String
showsPrec :: Int -> Signature -> ShowS
$cshowsPrec :: Int -> Signature -> ShowS
Show,ReadPrec [Signature]
ReadPrec Signature
Int -> ReadS Signature
ReadS [Signature]
forall a.
(Int -> ReadS a)
-> ReadS [a] -> ReadPrec a -> ReadPrec [a] -> Read a
readListPrec :: ReadPrec [Signature]
$creadListPrec :: ReadPrec [Signature]
readPrec :: ReadPrec Signature
$creadPrec :: ReadPrec Signature
readList :: ReadS [Signature]
$creadList :: ReadS [Signature]
readsPrec :: Int -> ReadS Signature
$creadsPrec :: Int -> ReadS Signature
Read,Signature -> Signature -> Bool
forall a. (a -> a -> Bool) -> (a -> a -> Bool) -> Eq a
/= :: Signature -> Signature -> Bool
$c/= :: Signature -> Signature -> Bool
== :: Signature -> Signature -> Bool
$c== :: Signature -> Signature -> Bool
Eq,Typeable Signature
Signature -> DataType
Signature -> Constr
(forall b. Data b => b -> b) -> Signature -> Signature
forall a.
Typeable a
-> (forall (c :: * -> *).
    (forall d b. Data d => c (d -> b) -> d -> c b)
    -> (forall g. g -> c g) -> a -> c a)
-> (forall (c :: * -> *).
    (forall b r. Data b => c (b -> r) -> c r)
    -> (forall r. r -> c r) -> Constr -> c a)
-> (a -> Constr)
-> (a -> DataType)
-> (forall (t :: * -> *) (c :: * -> *).
    Typeable t =>
    (forall d. Data d => c (t d)) -> Maybe (c a))
-> (forall (t :: * -> * -> *) (c :: * -> *).
    Typeable t =>
    (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c a))
-> ((forall b. Data b => b -> b) -> a -> a)
-> (forall r r'.
    (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> a -> r)
-> (forall r r'.
    (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> a -> r)
-> (forall u. (forall d. Data d => d -> u) -> a -> [u])
-> (forall u. Int -> (forall d. Data d => d -> u) -> a -> u)
-> (forall (m :: * -> *).
    Monad m =>
    (forall d. Data d => d -> m d) -> a -> m a)
-> (forall (m :: * -> *).
    MonadPlus m =>
    (forall d. Data d => d -> m d) -> a -> m a)
-> (forall (m :: * -> *).
    MonadPlus m =>
    (forall d. Data d => d -> m d) -> a -> m a)
-> Data a
forall u. Int -> (forall d. Data d => d -> u) -> Signature -> u
forall u. (forall d. Data d => d -> u) -> Signature -> [u]
forall r r'.
(r -> r' -> r)
-> r -> (forall d. Data d => d -> r') -> Signature -> r
forall r r'.
(r' -> r -> r)
-> r -> (forall d. Data d => d -> r') -> Signature -> r
forall (m :: * -> *).
Monad m =>
(forall d. Data d => d -> m d) -> Signature -> m Signature
forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> Signature -> m Signature
forall (c :: * -> *).
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c Signature
forall (c :: * -> *).
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> Signature -> c Signature
forall (t :: * -> *) (c :: * -> *).
Typeable t =>
(forall d. Data d => c (t d)) -> Maybe (c Signature)
forall (t :: * -> * -> *) (c :: * -> *).
Typeable t =>
(forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c Signature)
gmapMo :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> Signature -> m Signature
$cgmapMo :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> Signature -> m Signature
gmapMp :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> Signature -> m Signature
$cgmapMp :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> Signature -> m Signature
gmapM :: forall (m :: * -> *).
Monad m =>
(forall d. Data d => d -> m d) -> Signature -> m Signature
$cgmapM :: forall (m :: * -> *).
Monad m =>
(forall d. Data d => d -> m d) -> Signature -> m Signature
gmapQi :: forall u. Int -> (forall d. Data d => d -> u) -> Signature -> u
$cgmapQi :: forall u. Int -> (forall d. Data d => d -> u) -> Signature -> u
gmapQ :: forall u. (forall d. Data d => d -> u) -> Signature -> [u]
$cgmapQ :: forall u. (forall d. Data d => d -> u) -> Signature -> [u]
gmapQr :: forall r r'.
(r' -> r -> r)
-> r -> (forall d. Data d => d -> r') -> Signature -> r
$cgmapQr :: forall r r'.
(r' -> r -> r)
-> r -> (forall d. Data d => d -> r') -> Signature -> r
gmapQl :: forall r r'.
(r -> r' -> r)
-> r -> (forall d. Data d => d -> r') -> Signature -> r
$cgmapQl :: forall r r'.
(r -> r' -> r)
-> r -> (forall d. Data d => d -> r') -> Signature -> r
gmapT :: (forall b. Data b => b -> b) -> Signature -> Signature
$cgmapT :: (forall b. Data b => b -> b) -> Signature -> Signature
dataCast2 :: forall (t :: * -> * -> *) (c :: * -> *).
Typeable t =>
(forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c Signature)
$cdataCast2 :: forall (t :: * -> * -> *) (c :: * -> *).
Typeable t =>
(forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c Signature)
dataCast1 :: forall (t :: * -> *) (c :: * -> *).
Typeable t =>
(forall d. Data d => c (t d)) -> Maybe (c Signature)
$cdataCast1 :: forall (t :: * -> *) (c :: * -> *).
Typeable t =>
(forall d. Data d => c (t d)) -> Maybe (c Signature)
dataTypeOf :: Signature -> DataType
$cdataTypeOf :: Signature -> DataType
toConstr :: Signature -> Constr
$ctoConstr :: Signature -> Constr
gunfold :: forall (c :: * -> *).
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c Signature
$cgunfold :: forall (c :: * -> *).
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c Signature
gfoldl :: forall (c :: * -> *).
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> Signature -> c Signature
$cgfoldl :: forall (c :: * -> *).
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> Signature -> c Signature
Data)

instance NFData Signature where
    rnf :: Signature -> ()
rnf (Signature Integer
r Integer
s) = Integer
r seq :: forall a b. a -> b -> b
`seq` Integer
s seq :: forall a b. a -> b -> b
`seq` ()

-- | Represent a DSA public key.
data PublicKey = PublicKey
    { PublicKey -> Params
public_params :: Params       -- ^ DSA parameters
    , PublicKey -> Integer
public_y      :: PublicNumber -- ^ DSA public Y
    } deriving (Int -> PublicKey -> ShowS
[PublicKey] -> ShowS
PublicKey -> String
forall a.
(Int -> a -> ShowS) -> (a -> String) -> ([a] -> ShowS) -> Show a
showList :: [PublicKey] -> ShowS
$cshowList :: [PublicKey] -> ShowS
show :: PublicKey -> String
$cshow :: PublicKey -> String
showsPrec :: Int -> PublicKey -> ShowS
$cshowsPrec :: Int -> PublicKey -> ShowS
Show,ReadPrec [PublicKey]
ReadPrec PublicKey
Int -> ReadS PublicKey
ReadS [PublicKey]
forall a.
(Int -> ReadS a)
-> ReadS [a] -> ReadPrec a -> ReadPrec [a] -> Read a
readListPrec :: ReadPrec [PublicKey]
$creadListPrec :: ReadPrec [PublicKey]
readPrec :: ReadPrec PublicKey
$creadPrec :: ReadPrec PublicKey
readList :: ReadS [PublicKey]
$creadList :: ReadS [PublicKey]
readsPrec :: Int -> ReadS PublicKey
$creadsPrec :: Int -> ReadS PublicKey
Read,PublicKey -> PublicKey -> Bool
forall a. (a -> a -> Bool) -> (a -> a -> Bool) -> Eq a
/= :: PublicKey -> PublicKey -> Bool
$c/= :: PublicKey -> PublicKey -> Bool
== :: PublicKey -> PublicKey -> Bool
$c== :: PublicKey -> PublicKey -> Bool
Eq,Typeable PublicKey
PublicKey -> DataType
PublicKey -> Constr
(forall b. Data b => b -> b) -> PublicKey -> PublicKey
forall a.
Typeable a
-> (forall (c :: * -> *).
    (forall d b. Data d => c (d -> b) -> d -> c b)
    -> (forall g. g -> c g) -> a -> c a)
-> (forall (c :: * -> *).
    (forall b r. Data b => c (b -> r) -> c r)
    -> (forall r. r -> c r) -> Constr -> c a)
-> (a -> Constr)
-> (a -> DataType)
-> (forall (t :: * -> *) (c :: * -> *).
    Typeable t =>
    (forall d. Data d => c (t d)) -> Maybe (c a))
-> (forall (t :: * -> * -> *) (c :: * -> *).
    Typeable t =>
    (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c a))
-> ((forall b. Data b => b -> b) -> a -> a)
-> (forall r r'.
    (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> a -> r)
-> (forall r r'.
    (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> a -> r)
-> (forall u. (forall d. Data d => d -> u) -> a -> [u])
-> (forall u. Int -> (forall d. Data d => d -> u) -> a -> u)
-> (forall (m :: * -> *).
    Monad m =>
    (forall d. Data d => d -> m d) -> a -> m a)
-> (forall (m :: * -> *).
    MonadPlus m =>
    (forall d. Data d => d -> m d) -> a -> m a)
-> (forall (m :: * -> *).
    MonadPlus m =>
    (forall d. Data d => d -> m d) -> a -> m a)
-> Data a
forall u. Int -> (forall d. Data d => d -> u) -> PublicKey -> u
forall u. (forall d. Data d => d -> u) -> PublicKey -> [u]
forall r r'.
(r -> r' -> r)
-> r -> (forall d. Data d => d -> r') -> PublicKey -> r
forall r r'.
(r' -> r -> r)
-> r -> (forall d. Data d => d -> r') -> PublicKey -> r
forall (m :: * -> *).
Monad m =>
(forall d. Data d => d -> m d) -> PublicKey -> m PublicKey
forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> PublicKey -> m PublicKey
forall (c :: * -> *).
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c PublicKey
forall (c :: * -> *).
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> PublicKey -> c PublicKey
forall (t :: * -> *) (c :: * -> *).
Typeable t =>
(forall d. Data d => c (t d)) -> Maybe (c PublicKey)
forall (t :: * -> * -> *) (c :: * -> *).
Typeable t =>
(forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c PublicKey)
gmapMo :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> PublicKey -> m PublicKey
$cgmapMo :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> PublicKey -> m PublicKey
gmapMp :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> PublicKey -> m PublicKey
$cgmapMp :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> PublicKey -> m PublicKey
gmapM :: forall (m :: * -> *).
Monad m =>
(forall d. Data d => d -> m d) -> PublicKey -> m PublicKey
$cgmapM :: forall (m :: * -> *).
Monad m =>
(forall d. Data d => d -> m d) -> PublicKey -> m PublicKey
gmapQi :: forall u. Int -> (forall d. Data d => d -> u) -> PublicKey -> u
$cgmapQi :: forall u. Int -> (forall d. Data d => d -> u) -> PublicKey -> u
gmapQ :: forall u. (forall d. Data d => d -> u) -> PublicKey -> [u]
$cgmapQ :: forall u. (forall d. Data d => d -> u) -> PublicKey -> [u]
gmapQr :: forall r r'.
(r' -> r -> r)
-> r -> (forall d. Data d => d -> r') -> PublicKey -> r
$cgmapQr :: forall r r'.
(r' -> r -> r)
-> r -> (forall d. Data d => d -> r') -> PublicKey -> r
gmapQl :: forall r r'.
(r -> r' -> r)
-> r -> (forall d. Data d => d -> r') -> PublicKey -> r
$cgmapQl :: forall r r'.
(r -> r' -> r)
-> r -> (forall d. Data d => d -> r') -> PublicKey -> r
gmapT :: (forall b. Data b => b -> b) -> PublicKey -> PublicKey
$cgmapT :: (forall b. Data b => b -> b) -> PublicKey -> PublicKey
dataCast2 :: forall (t :: * -> * -> *) (c :: * -> *).
Typeable t =>
(forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c PublicKey)
$cdataCast2 :: forall (t :: * -> * -> *) (c :: * -> *).
Typeable t =>
(forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c PublicKey)
dataCast1 :: forall (t :: * -> *) (c :: * -> *).
Typeable t =>
(forall d. Data d => c (t d)) -> Maybe (c PublicKey)
$cdataCast1 :: forall (t :: * -> *) (c :: * -> *).
Typeable t =>
(forall d. Data d => c (t d)) -> Maybe (c PublicKey)
dataTypeOf :: PublicKey -> DataType
$cdataTypeOf :: PublicKey -> DataType
toConstr :: PublicKey -> Constr
$ctoConstr :: PublicKey -> Constr
gunfold :: forall (c :: * -> *).
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c PublicKey
$cgunfold :: forall (c :: * -> *).
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c PublicKey
gfoldl :: forall (c :: * -> *).
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> PublicKey -> c PublicKey
$cgfoldl :: forall (c :: * -> *).
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> PublicKey -> c PublicKey
Data)

instance NFData PublicKey where
    rnf :: PublicKey -> ()
rnf (PublicKey Params
params Integer
y) = Integer
y seq :: forall a b. a -> b -> b
`seq` Params
params seq :: forall a b. a -> b -> b
`seq` ()

-- | Represent a DSA private key.
--
-- Only x need to be secret.
-- the DSA parameters are publicly shared with the other side.
data PrivateKey = PrivateKey
    { PrivateKey -> Params
private_params :: Params        -- ^ DSA parameters
    , PrivateKey -> Integer
private_x      :: PrivateNumber -- ^ DSA private X
    } deriving (Int -> PrivateKey -> ShowS
[PrivateKey] -> ShowS
PrivateKey -> String
forall a.
(Int -> a -> ShowS) -> (a -> String) -> ([a] -> ShowS) -> Show a
showList :: [PrivateKey] -> ShowS
$cshowList :: [PrivateKey] -> ShowS
show :: PrivateKey -> String
$cshow :: PrivateKey -> String
showsPrec :: Int -> PrivateKey -> ShowS
$cshowsPrec :: Int -> PrivateKey -> ShowS
Show,ReadPrec [PrivateKey]
ReadPrec PrivateKey
Int -> ReadS PrivateKey
ReadS [PrivateKey]
forall a.
(Int -> ReadS a)
-> ReadS [a] -> ReadPrec a -> ReadPrec [a] -> Read a
readListPrec :: ReadPrec [PrivateKey]
$creadListPrec :: ReadPrec [PrivateKey]
readPrec :: ReadPrec PrivateKey
$creadPrec :: ReadPrec PrivateKey
readList :: ReadS [PrivateKey]
$creadList :: ReadS [PrivateKey]
readsPrec :: Int -> ReadS PrivateKey
$creadsPrec :: Int -> ReadS PrivateKey
Read,PrivateKey -> PrivateKey -> Bool
forall a. (a -> a -> Bool) -> (a -> a -> Bool) -> Eq a
/= :: PrivateKey -> PrivateKey -> Bool
$c/= :: PrivateKey -> PrivateKey -> Bool
== :: PrivateKey -> PrivateKey -> Bool
$c== :: PrivateKey -> PrivateKey -> Bool
Eq,Typeable PrivateKey
PrivateKey -> DataType
PrivateKey -> Constr
(forall b. Data b => b -> b) -> PrivateKey -> PrivateKey
forall a.
Typeable a
-> (forall (c :: * -> *).
    (forall d b. Data d => c (d -> b) -> d -> c b)
    -> (forall g. g -> c g) -> a -> c a)
-> (forall (c :: * -> *).
    (forall b r. Data b => c (b -> r) -> c r)
    -> (forall r. r -> c r) -> Constr -> c a)
-> (a -> Constr)
-> (a -> DataType)
-> (forall (t :: * -> *) (c :: * -> *).
    Typeable t =>
    (forall d. Data d => c (t d)) -> Maybe (c a))
-> (forall (t :: * -> * -> *) (c :: * -> *).
    Typeable t =>
    (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c a))
-> ((forall b. Data b => b -> b) -> a -> a)
-> (forall r r'.
    (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> a -> r)
-> (forall r r'.
    (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> a -> r)
-> (forall u. (forall d. Data d => d -> u) -> a -> [u])
-> (forall u. Int -> (forall d. Data d => d -> u) -> a -> u)
-> (forall (m :: * -> *).
    Monad m =>
    (forall d. Data d => d -> m d) -> a -> m a)
-> (forall (m :: * -> *).
    MonadPlus m =>
    (forall d. Data d => d -> m d) -> a -> m a)
-> (forall (m :: * -> *).
    MonadPlus m =>
    (forall d. Data d => d -> m d) -> a -> m a)
-> Data a
forall u. Int -> (forall d. Data d => d -> u) -> PrivateKey -> u
forall u. (forall d. Data d => d -> u) -> PrivateKey -> [u]
forall r r'.
(r -> r' -> r)
-> r -> (forall d. Data d => d -> r') -> PrivateKey -> r
forall r r'.
(r' -> r -> r)
-> r -> (forall d. Data d => d -> r') -> PrivateKey -> r
forall (m :: * -> *).
Monad m =>
(forall d. Data d => d -> m d) -> PrivateKey -> m PrivateKey
forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> PrivateKey -> m PrivateKey
forall (c :: * -> *).
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c PrivateKey
forall (c :: * -> *).
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> PrivateKey -> c PrivateKey
forall (t :: * -> *) (c :: * -> *).
Typeable t =>
(forall d. Data d => c (t d)) -> Maybe (c PrivateKey)
forall (t :: * -> * -> *) (c :: * -> *).
Typeable t =>
(forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c PrivateKey)
gmapMo :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> PrivateKey -> m PrivateKey
$cgmapMo :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> PrivateKey -> m PrivateKey
gmapMp :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> PrivateKey -> m PrivateKey
$cgmapMp :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> PrivateKey -> m PrivateKey
gmapM :: forall (m :: * -> *).
Monad m =>
(forall d. Data d => d -> m d) -> PrivateKey -> m PrivateKey
$cgmapM :: forall (m :: * -> *).
Monad m =>
(forall d. Data d => d -> m d) -> PrivateKey -> m PrivateKey
gmapQi :: forall u. Int -> (forall d. Data d => d -> u) -> PrivateKey -> u
$cgmapQi :: forall u. Int -> (forall d. Data d => d -> u) -> PrivateKey -> u
gmapQ :: forall u. (forall d. Data d => d -> u) -> PrivateKey -> [u]
$cgmapQ :: forall u. (forall d. Data d => d -> u) -> PrivateKey -> [u]
gmapQr :: forall r r'.
(r' -> r -> r)
-> r -> (forall d. Data d => d -> r') -> PrivateKey -> r
$cgmapQr :: forall r r'.
(r' -> r -> r)
-> r -> (forall d. Data d => d -> r') -> PrivateKey -> r
gmapQl :: forall r r'.
(r -> r' -> r)
-> r -> (forall d. Data d => d -> r') -> PrivateKey -> r
$cgmapQl :: forall r r'.
(r -> r' -> r)
-> r -> (forall d. Data d => d -> r') -> PrivateKey -> r
gmapT :: (forall b. Data b => b -> b) -> PrivateKey -> PrivateKey
$cgmapT :: (forall b. Data b => b -> b) -> PrivateKey -> PrivateKey
dataCast2 :: forall (t :: * -> * -> *) (c :: * -> *).
Typeable t =>
(forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c PrivateKey)
$cdataCast2 :: forall (t :: * -> * -> *) (c :: * -> *).
Typeable t =>
(forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c PrivateKey)
dataCast1 :: forall (t :: * -> *) (c :: * -> *).
Typeable t =>
(forall d. Data d => c (t d)) -> Maybe (c PrivateKey)
$cdataCast1 :: forall (t :: * -> *) (c :: * -> *).
Typeable t =>
(forall d. Data d => c (t d)) -> Maybe (c PrivateKey)
dataTypeOf :: PrivateKey -> DataType
$cdataTypeOf :: PrivateKey -> DataType
toConstr :: PrivateKey -> Constr
$ctoConstr :: PrivateKey -> Constr
gunfold :: forall (c :: * -> *).
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c PrivateKey
$cgunfold :: forall (c :: * -> *).
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c PrivateKey
gfoldl :: forall (c :: * -> *).
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> PrivateKey -> c PrivateKey
$cgfoldl :: forall (c :: * -> *).
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> PrivateKey -> c PrivateKey
Data)

instance NFData PrivateKey where
    rnf :: PrivateKey -> ()
rnf (PrivateKey Params
params Integer
x) = Integer
x seq :: forall a b. a -> b -> b
`seq` Params
params seq :: forall a b. a -> b -> b
`seq` ()

-- | Represent a DSA key pair
data KeyPair = KeyPair Params PublicNumber PrivateNumber
    deriving (Int -> KeyPair -> ShowS
[KeyPair] -> ShowS
KeyPair -> String
forall a.
(Int -> a -> ShowS) -> (a -> String) -> ([a] -> ShowS) -> Show a
showList :: [KeyPair] -> ShowS
$cshowList :: [KeyPair] -> ShowS
show :: KeyPair -> String
$cshow :: KeyPair -> String
showsPrec :: Int -> KeyPair -> ShowS
$cshowsPrec :: Int -> KeyPair -> ShowS
Show,ReadPrec [KeyPair]
ReadPrec KeyPair
Int -> ReadS KeyPair
ReadS [KeyPair]
forall a.
(Int -> ReadS a)
-> ReadS [a] -> ReadPrec a -> ReadPrec [a] -> Read a
readListPrec :: ReadPrec [KeyPair]
$creadListPrec :: ReadPrec [KeyPair]
readPrec :: ReadPrec KeyPair
$creadPrec :: ReadPrec KeyPair
readList :: ReadS [KeyPair]
$creadList :: ReadS [KeyPair]
readsPrec :: Int -> ReadS KeyPair
$creadsPrec :: Int -> ReadS KeyPair
Read,KeyPair -> KeyPair -> Bool
forall a. (a -> a -> Bool) -> (a -> a -> Bool) -> Eq a
/= :: KeyPair -> KeyPair -> Bool
$c/= :: KeyPair -> KeyPair -> Bool
== :: KeyPair -> KeyPair -> Bool
$c== :: KeyPair -> KeyPair -> Bool
Eq,Typeable KeyPair
KeyPair -> DataType
KeyPair -> Constr
(forall b. Data b => b -> b) -> KeyPair -> KeyPair
forall a.
Typeable a
-> (forall (c :: * -> *).
    (forall d b. Data d => c (d -> b) -> d -> c b)
    -> (forall g. g -> c g) -> a -> c a)
-> (forall (c :: * -> *).
    (forall b r. Data b => c (b -> r) -> c r)
    -> (forall r. r -> c r) -> Constr -> c a)
-> (a -> Constr)
-> (a -> DataType)
-> (forall (t :: * -> *) (c :: * -> *).
    Typeable t =>
    (forall d. Data d => c (t d)) -> Maybe (c a))
-> (forall (t :: * -> * -> *) (c :: * -> *).
    Typeable t =>
    (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c a))
-> ((forall b. Data b => b -> b) -> a -> a)
-> (forall r r'.
    (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> a -> r)
-> (forall r r'.
    (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> a -> r)
-> (forall u. (forall d. Data d => d -> u) -> a -> [u])
-> (forall u. Int -> (forall d. Data d => d -> u) -> a -> u)
-> (forall (m :: * -> *).
    Monad m =>
    (forall d. Data d => d -> m d) -> a -> m a)
-> (forall (m :: * -> *).
    MonadPlus m =>
    (forall d. Data d => d -> m d) -> a -> m a)
-> (forall (m :: * -> *).
    MonadPlus m =>
    (forall d. Data d => d -> m d) -> a -> m a)
-> Data a
forall u. Int -> (forall d. Data d => d -> u) -> KeyPair -> u
forall u. (forall d. Data d => d -> u) -> KeyPair -> [u]
forall r r'.
(r -> r' -> r)
-> r -> (forall d. Data d => d -> r') -> KeyPair -> r
forall r r'.
(r' -> r -> r)
-> r -> (forall d. Data d => d -> r') -> KeyPair -> r
forall (m :: * -> *).
Monad m =>
(forall d. Data d => d -> m d) -> KeyPair -> m KeyPair
forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> KeyPair -> m KeyPair
forall (c :: * -> *).
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c KeyPair
forall (c :: * -> *).
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> KeyPair -> c KeyPair
forall (t :: * -> *) (c :: * -> *).
Typeable t =>
(forall d. Data d => c (t d)) -> Maybe (c KeyPair)
forall (t :: * -> * -> *) (c :: * -> *).
Typeable t =>
(forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c KeyPair)
gmapMo :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> KeyPair -> m KeyPair
$cgmapMo :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> KeyPair -> m KeyPair
gmapMp :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> KeyPair -> m KeyPair
$cgmapMp :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> KeyPair -> m KeyPair
gmapM :: forall (m :: * -> *).
Monad m =>
(forall d. Data d => d -> m d) -> KeyPair -> m KeyPair
$cgmapM :: forall (m :: * -> *).
Monad m =>
(forall d. Data d => d -> m d) -> KeyPair -> m KeyPair
gmapQi :: forall u. Int -> (forall d. Data d => d -> u) -> KeyPair -> u
$cgmapQi :: forall u. Int -> (forall d. Data d => d -> u) -> KeyPair -> u
gmapQ :: forall u. (forall d. Data d => d -> u) -> KeyPair -> [u]
$cgmapQ :: forall u. (forall d. Data d => d -> u) -> KeyPair -> [u]
gmapQr :: forall r r'.
(r' -> r -> r)
-> r -> (forall d. Data d => d -> r') -> KeyPair -> r
$cgmapQr :: forall r r'.
(r' -> r -> r)
-> r -> (forall d. Data d => d -> r') -> KeyPair -> r
gmapQl :: forall r r'.
(r -> r' -> r)
-> r -> (forall d. Data d => d -> r') -> KeyPair -> r
$cgmapQl :: forall r r'.
(r -> r' -> r)
-> r -> (forall d. Data d => d -> r') -> KeyPair -> r
gmapT :: (forall b. Data b => b -> b) -> KeyPair -> KeyPair
$cgmapT :: (forall b. Data b => b -> b) -> KeyPair -> KeyPair
dataCast2 :: forall (t :: * -> * -> *) (c :: * -> *).
Typeable t =>
(forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c KeyPair)
$cdataCast2 :: forall (t :: * -> * -> *) (c :: * -> *).
Typeable t =>
(forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c KeyPair)
dataCast1 :: forall (t :: * -> *) (c :: * -> *).
Typeable t =>
(forall d. Data d => c (t d)) -> Maybe (c KeyPair)
$cdataCast1 :: forall (t :: * -> *) (c :: * -> *).
Typeable t =>
(forall d. Data d => c (t d)) -> Maybe (c KeyPair)
dataTypeOf :: KeyPair -> DataType
$cdataTypeOf :: KeyPair -> DataType
toConstr :: KeyPair -> Constr
$ctoConstr :: KeyPair -> Constr
gunfold :: forall (c :: * -> *).
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c KeyPair
$cgunfold :: forall (c :: * -> *).
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c KeyPair
gfoldl :: forall (c :: * -> *).
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> KeyPair -> c KeyPair
$cgfoldl :: forall (c :: * -> *).
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> KeyPair -> c KeyPair
Data)

instance NFData KeyPair where
    rnf :: KeyPair -> ()
rnf (KeyPair Params
params Integer
y Integer
x) = Integer
x seq :: forall a b. a -> b -> b
`seq` Integer
y seq :: forall a b. a -> b -> b
`seq` Params
params seq :: forall a b. a -> b -> b
`seq` ()

-- | Public key of a DSA Key pair
toPublicKey :: KeyPair -> PublicKey
toPublicKey :: KeyPair -> PublicKey
toPublicKey (KeyPair Params
params Integer
pub Integer
_) = Params -> Integer -> PublicKey
PublicKey Params
params Integer
pub

-- | Private key of a DSA Key pair
toPrivateKey :: KeyPair -> PrivateKey
toPrivateKey :: KeyPair -> PrivateKey
toPrivateKey (KeyPair Params
params Integer
_ Integer
priv) = Params -> Integer -> PrivateKey
PrivateKey Params
params Integer
priv

-- | generate a private number with no specific property
-- this number is usually called X in DSA text.
generatePrivate :: MonadRandom m => Params -> m PrivateNumber
generatePrivate :: forall (m :: * -> *). MonadRandom m => Params -> m Integer
generatePrivate (Params Integer
_ Integer
_ Integer
q) = forall (m :: * -> *). MonadRandom m => Integer -> m Integer
generateMax Integer
q

-- | Calculate the public number from the parameters and the private key
calculatePublic :: Params -> PrivateNumber -> PublicNumber
calculatePublic :: Params -> Integer -> Integer
calculatePublic (Params Integer
p Integer
g Integer
_) Integer
x = Integer -> Integer -> Integer -> Integer
expSafe Integer
g Integer
x Integer
p

-- | sign message using the private key and an explicit k number.
signWith :: (ByteArrayAccess msg, HashAlgorithm hash)
         => Integer         -- ^ k random number
         -> PrivateKey      -- ^ private key
         -> hash            -- ^ hash function
         -> msg             -- ^ message to sign
         -> Maybe Signature
signWith :: forall msg hash.
(ByteArrayAccess msg, HashAlgorithm hash) =>
Integer -> PrivateKey -> hash -> msg -> Maybe Signature
signWith Integer
k PrivateKey
pk hash
hashAlg msg
msg
    | Integer
r forall a. Eq a => a -> a -> Bool
== Integer
0 Bool -> Bool -> Bool
|| Integer
s forall a. Eq a => a -> a -> Bool
== Integer
0  = forall a. Maybe a
Nothing
    | Bool
otherwise         = forall a. a -> Maybe a
Just forall a b. (a -> b) -> a -> b
$ Integer -> Integer -> Signature
Signature Integer
r Integer
s
    where -- parameters
          (Params Integer
p Integer
g Integer
q) = PrivateKey -> Params
private_params PrivateKey
pk
          x :: Integer
x              = PrivateKey -> Integer
private_x PrivateKey
pk
          -- compute r,s
          kInv :: Integer
kInv      = forall a. HasCallStack => Maybe a -> a
fromJust forall a b. (a -> b) -> a -> b
$ Integer -> Integer -> Maybe Integer
inverse Integer
k Integer
q
          hm :: Integer
hm        = forall msg hash.
(ByteArrayAccess msg, HashAlgorithm hash) =>
hash -> msg -> Integer -> Integer
dsaTruncHash hash
hashAlg msg
msg Integer
q
          r :: Integer
r         = Integer -> Integer -> Integer -> Integer
expSafe Integer
g Integer
k Integer
p forall a. Integral a => a -> a -> a
`mod` Integer
q
          s :: Integer
s         = (Integer
kInv forall a. Num a => a -> a -> a
* (Integer
hm forall a. Num a => a -> a -> a
+ Integer
x forall a. Num a => a -> a -> a
* Integer
r)) forall a. Integral a => a -> a -> a
`mod` Integer
q

-- | sign message using the private key.
sign :: (ByteArrayAccess msg, HashAlgorithm hash, MonadRandom m) => PrivateKey -> hash -> msg -> m Signature
sign :: forall msg hash (m :: * -> *).
(ByteArrayAccess msg, HashAlgorithm hash, MonadRandom m) =>
PrivateKey -> hash -> msg -> m Signature
sign PrivateKey
pk hash
hashAlg msg
msg = do
    Integer
k <- forall (m :: * -> *). MonadRandom m => Integer -> m Integer
generateMax Integer
q
    case forall msg hash.
(ByteArrayAccess msg, HashAlgorithm hash) =>
Integer -> PrivateKey -> hash -> msg -> Maybe Signature
signWith Integer
k PrivateKey
pk hash
hashAlg msg
msg of
        Maybe Signature
Nothing  -> forall msg hash (m :: * -> *).
(ByteArrayAccess msg, HashAlgorithm hash, MonadRandom m) =>
PrivateKey -> hash -> msg -> m Signature
sign PrivateKey
pk hash
hashAlg msg
msg
        Just Signature
sig -> forall (m :: * -> *) a. Monad m => a -> m a
return Signature
sig
  where
    (Params Integer
_ Integer
_ Integer
q) = PrivateKey -> Params
private_params PrivateKey
pk

-- | verify a bytestring using the public key.
verify :: (ByteArrayAccess msg, HashAlgorithm hash) => hash -> PublicKey -> Signature -> msg -> Bool
verify :: forall msg hash.
(ByteArrayAccess msg, HashAlgorithm hash) =>
hash -> PublicKey -> Signature -> msg -> Bool
verify hash
hashAlg PublicKey
pk (Signature Integer
r Integer
s) msg
m
    -- Reject the signature if either 0 < r < q or 0 < s < q is not satisfied.
    | Integer
r forall a. Ord a => a -> a -> Bool
<= Integer
0 Bool -> Bool -> Bool
|| Integer
r forall a. Ord a => a -> a -> Bool
>= Integer
q Bool -> Bool -> Bool
|| Integer
s forall a. Ord a => a -> a -> Bool
<= Integer
0 Bool -> Bool -> Bool
|| Integer
s forall a. Ord a => a -> a -> Bool
>= Integer
q = Bool
False
    | Bool
otherwise                            = Integer
v forall a. Eq a => a -> a -> Bool
== Integer
r
    where (Params Integer
p Integer
g Integer
q) = PublicKey -> Params
public_params PublicKey
pk
          y :: Integer
y       = PublicKey -> Integer
public_y PublicKey
pk
          hm :: Integer
hm      = forall msg hash.
(ByteArrayAccess msg, HashAlgorithm hash) =>
hash -> msg -> Integer -> Integer
dsaTruncHash hash
hashAlg msg
m Integer
q
          w :: Integer
w       = forall a. HasCallStack => Maybe a -> a
fromJust forall a b. (a -> b) -> a -> b
$ Integer -> Integer -> Maybe Integer
inverse Integer
s Integer
q
          u1 :: Integer
u1      = (Integer
hmforall a. Num a => a -> a -> a
*Integer
w) forall a. Integral a => a -> a -> a
`mod` Integer
q
          u2 :: Integer
u2      = (Integer
rforall a. Num a => a -> a -> a
*Integer
w) forall a. Integral a => a -> a -> a
`mod` Integer
q
          v :: Integer
v       = ((Integer -> Integer -> Integer -> Integer
expFast Integer
g Integer
u1 Integer
p) forall a. Num a => a -> a -> a
* (Integer -> Integer -> Integer -> Integer
expFast Integer
y Integer
u2 Integer
p)) forall a. Integral a => a -> a -> a
`mod` Integer
p forall a. Integral a => a -> a -> a
`mod` Integer
q