-- |
-- Module      : Crypto.PubKey.Rabin.Basic
-- License     : BSD-style
-- Maintainer  : Carlos Rodriguez-Vega <crodveg@yahoo.es>
-- Stability   : experimental
-- Portability : unknown
--
-- Rabin cryptosystem for public-key cryptography and digital signature.
--
{-# LANGUAGE DeriveDataTypeable #-}
module Crypto.PubKey.Rabin.Basic
    ( PublicKey(..)
    , PrivateKey(..)
    , Signature(..)
    , generate
    , encrypt
    , encryptWithSeed
    , decrypt
    , sign
    , signWith
    , verify
    ) where

import           Data.ByteString (ByteString)
import qualified Data.ByteString as B
import           Data.Data
import           Data.Either (rights)

import           Crypto.Hash
import           Crypto.Number.Basic (gcde, numBytes)
import           Crypto.Number.ModArithmetic (expSafe, jacobi)
import           Crypto.Number.Serialize (i2osp, i2ospOf_, os2ip)
import           Crypto.PubKey.Rabin.OAEP 
import           Crypto.PubKey.Rabin.Types
import           Crypto.Random (MonadRandom, getRandomBytes)

-- | Represent a Rabin public key.
data PublicKey = PublicKey
    { PublicKey -> Int
public_size :: Int      -- ^ size of key in bytes
    , PublicKey -> Integer
public_n    :: Integer  -- ^ public p*q
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-- | Represent a Rabin private key.
data PrivateKey = PrivateKey
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-- | Rabin Signature.
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-- | Generate a pair of (private, public) key of size in bytes.
-- Primes p and q are both congruent 3 mod 4.
--
-- See algorithm 8.11 in "Handbook of Applied Cryptography" by Alfred J. Menezes et al.
generate :: MonadRandom m
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q
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q
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n }
            privateKey :: PrivateKey
privateKey = PrivateKey { private_pub :: PublicKey
private_pub = PublicKey
publicKey
                                    , private_p :: Integer
private_p   = Integer
p
                                    , private_q :: Integer
private_q   = Integer
q
                                    , private_a :: Integer
private_a   = Integer
a
                                    , private_b :: Integer
private_b   = Integer
b }
            in (PublicKey
publicKey, PrivateKey
privateKey)

-- | Encrypt plaintext using public key an a predefined OAEP seed.
--
-- See algorithm 8.11 in "Handbook of Applied Cryptography" by Alfred J. Menezes et al.
encryptWithSeed :: HashAlgorithm hash
                => ByteString                               -- ^ Seed
                -> OAEPParams hash ByteString ByteString    -- ^ OAEP padding
                -> PublicKey                                -- ^ public key
                -> ByteString                               -- ^ plaintext
                -> Either Error ByteString
encryptWithSeed :: forall hash.
HashAlgorithm hash =>
ByteString
-> OAEPParams hash ByteString ByteString
-> PublicKey
-> ByteString
-> Either Error ByteString
encryptWithSeed ByteString
seed OAEPParams hash ByteString ByteString
oaep PublicKey
pk ByteString
m =
    let n :: Integer
n  = PublicKey -> Integer
public_n PublicKey
pk
        k :: Int
k  = Integer -> Int
numBytes Integer
n
     in do
        ByteString
m' <- forall hash.
HashAlgorithm hash =>
ByteString
-> OAEPParams hash ByteString ByteString
-> Int
-> ByteString
-> Either Error ByteString
pad ByteString
seed OAEPParams hash ByteString ByteString
oaep Int
k ByteString
m
        let m'' :: Integer
m'' = forall ba. ByteArrayAccess ba => ba -> Integer
os2ip ByteString
m'
        forall (m :: * -> *) a. Monad m => a -> m a
return forall a b. (a -> b) -> a -> b
$ forall ba. ByteArray ba => Integer -> ba
i2osp forall a b. (a -> b) -> a -> b
$ Integer -> Integer -> Integer -> Integer
expSafe Integer
m'' Integer
2 Integer
n

-- | Encrypt plaintext using public key.
encrypt :: (HashAlgorithm hash, MonadRandom m)
        => OAEPParams hash ByteString ByteString    -- ^ OAEP padding parameters
        -> PublicKey                                -- ^ public key
        -> ByteString                               -- ^ plaintext 
        -> m (Either Error ByteString)
encrypt :: forall hash (m :: * -> *).
(HashAlgorithm hash, MonadRandom m) =>
OAEPParams hash ByteString ByteString
-> PublicKey -> ByteString -> m (Either Error ByteString)
encrypt OAEPParams hash ByteString ByteString
oaep PublicKey
pk ByteString
m = do
    ByteString
seed <- forall (m :: * -> *) byteArray.
(MonadRandom m, ByteArray byteArray) =>
Int -> m byteArray
getRandomBytes Int
hashLen
    forall (m :: * -> *) a. Monad m => a -> m a
return forall a b. (a -> b) -> a -> b
$ forall hash.
HashAlgorithm hash =>
ByteString
-> OAEPParams hash ByteString ByteString
-> PublicKey
-> ByteString
-> Either Error ByteString
encryptWithSeed ByteString
seed OAEPParams hash ByteString ByteString
oaep PublicKey
pk ByteString
m
  where
    hashLen :: Int
hashLen = forall a. HashAlgorithm a => a -> Int
hashDigestSize (forall hash seed output. OAEPParams hash seed output -> hash
oaepHash OAEPParams hash ByteString ByteString
oaep) 

-- | Decrypt ciphertext using private key.
--
-- See algorithm 8.12 in "Handbook of Applied Cryptography" by Alfred J. Menezes et al.
decrypt :: HashAlgorithm hash
        => OAEPParams hash ByteString ByteString    -- ^ OAEP padding parameters
        -> PrivateKey                               -- ^ private key
        -> ByteString                               -- ^ ciphertext
        -> Maybe ByteString
decrypt :: forall hash.
HashAlgorithm hash =>
OAEPParams hash ByteString ByteString
-> PrivateKey -> ByteString -> Maybe ByteString
decrypt OAEPParams hash ByteString ByteString
oaep PrivateKey
pk ByteString
c =
    let p :: Integer
p  = PrivateKey -> Integer
private_p PrivateKey
pk 
        q :: Integer
q  = PrivateKey -> Integer
private_q PrivateKey
pk     
        a :: Integer
a  = PrivateKey -> Integer
private_a PrivateKey
pk 
        b :: Integer
b  = PrivateKey -> Integer
private_b PrivateKey
pk
        n :: Integer
n  = PublicKey -> Integer
public_n forall a b. (a -> b) -> a -> b
$ PrivateKey -> PublicKey
private_pub PrivateKey
pk
        k :: Int
k  = Integer -> Int
numBytes Integer
n
        c' :: Integer
c' = forall ba. ByteArrayAccess ba => ba -> Integer
os2ip ByteString
c
        solutions :: [ByteString]
solutions = forall a b. [Either a b] -> [b]
rights forall a b. (a -> b) -> a -> b
$ forall {a}. (a, a, a, a) -> [a]
toList forall a b. (a -> b) -> a -> b
$ forall {t} {d}. (t -> d) -> (t, t, t, t) -> (d, d, d, d)
mapTuple (forall hash.
HashAlgorithm hash =>
OAEPParams hash ByteString ByteString
-> Int -> ByteString -> Either Error ByteString
unpad OAEPParams hash ByteString ByteString
oaep Int
k forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall ba. ByteArray ba => Int -> Integer -> ba
i2ospOf_ Int
k) forall a b. (a -> b) -> a -> b
$ Integer
-> Integer
-> Integer
-> Integer
-> Integer
-> Integer
-> (Integer, Integer, Integer, Integer)
sqroot' Integer
c' Integer
p Integer
q Integer
a Integer
b Integer
n
     in if forall (t :: * -> *) a. Foldable t => t a -> Int
length [ByteString]
solutions forall a. Eq a => a -> a -> Bool
/= Int
1 then forall a. Maybe a
Nothing
        else forall a. a -> Maybe a
Just forall a b. (a -> b) -> a -> b
$ forall a. [a] -> a
head [ByteString]
solutions
      where toList :: (a, a, a, a) -> [a]
toList (a
w, a
x, a
y, a
z) = a
wforall a. a -> [a] -> [a]
:a
xforall a. a -> [a] -> [a]
:a
yforall a. a -> [a] -> [a]
:a
zforall a. a -> [a] -> [a]
:[]
            mapTuple :: (t -> d) -> (t, t, t, t) -> (d, d, d, d)
mapTuple t -> d
f (t
w, t
x, t
y, t
z) = (t -> d
f t
w, t -> d
f t
x, t -> d
f t
y, t -> d
f t
z)

-- | Sign message using padding, hash algorithm and private key.
--
-- See <https://en.wikipedia.org/wiki/Rabin_signature_algorithm>.
signWith :: HashAlgorithm hash
         => ByteString    -- ^ padding
         -> PrivateKey    -- ^ private key
         -> hash          -- ^ hash function
         -> ByteString    -- ^ message to sign
         -> Either Error Signature
signWith :: forall hash.
HashAlgorithm hash =>
ByteString
-> PrivateKey -> hash -> ByteString -> Either Error Signature
signWith ByteString
padding PrivateKey
pk hash
hashAlg ByteString
m = do
    Integer
h <- forall hash.
HashAlgorithm hash =>
ByteString
-> PrivateKey -> hash -> ByteString -> Either Error Integer
calculateHash ByteString
padding PrivateKey
pk hash
hashAlg ByteString
m
    Signature
signature <- Integer -> Either Error Signature
calculateSignature Integer
h
    forall (m :: * -> *) a. Monad m => a -> m a
return Signature
signature
  where
    calculateSignature :: Integer -> Either Error Signature
calculateSignature Integer
h =
        let p :: Integer
p = PrivateKey -> Integer
private_p PrivateKey
pk
            q :: Integer
q = PrivateKey -> Integer
private_q PrivateKey
pk     
            a :: Integer
a = PrivateKey -> Integer
private_a PrivateKey
pk 
            b :: Integer
b = PrivateKey -> Integer
private_b PrivateKey
pk
            n :: Integer
n = PublicKey -> Integer
public_n forall a b. (a -> b) -> a -> b
$ PrivateKey -> PublicKey
private_pub PrivateKey
pk
         in if Integer
h forall a. Ord a => a -> a -> Bool
>= Integer
n then forall a b. a -> Either a b
Left Error
MessageTooLong
            else let (Integer
r, Integer
_, Integer
_, Integer
_) = Integer
-> Integer
-> Integer
-> Integer
-> Integer
-> Integer
-> (Integer, Integer, Integer, Integer)
sqroot' Integer
h Integer
p Integer
q Integer
a Integer
b Integer
n
                  in forall a b. b -> Either a b
Right forall a b. (a -> b) -> a -> b
$ (Integer, Integer) -> Signature
Signature (forall ba. ByteArrayAccess ba => ba -> Integer
os2ip ByteString
padding, Integer
r)

-- | Sign message using hash algorithm and private key.
--
-- See <https://en.wikipedia.org/wiki/Rabin_signature_algorithm>.
sign :: (MonadRandom m, HashAlgorithm hash)
     => PrivateKey    -- ^ private key
     -> hash          -- ^ hash function
     -> ByteString    -- ^ message to sign
     -> m (Either Error Signature)
sign :: forall (m :: * -> *) hash.
(MonadRandom m, HashAlgorithm hash) =>
PrivateKey -> hash -> ByteString -> m (Either Error Signature)
sign PrivateKey
pk hash
hashAlg ByteString
m = do
    ByteString
padding <- m ByteString
findPadding
    forall (m :: * -> *) a. Monad m => a -> m a
return forall a b. (a -> b) -> a -> b
$ forall hash.
HashAlgorithm hash =>
ByteString
-> PrivateKey -> hash -> ByteString -> Either Error Signature
signWith ByteString
padding PrivateKey
pk hash
hashAlg ByteString
m
  where 
    findPadding :: m ByteString
findPadding = do
        ByteString
padding <- forall (m :: * -> *) byteArray.
(MonadRandom m, ByteArray byteArray) =>
Int -> m byteArray
getRandomBytes Int
8
        case forall hash.
HashAlgorithm hash =>
ByteString
-> PrivateKey -> hash -> ByteString -> Either Error Integer
calculateHash ByteString
padding PrivateKey
pk hash
hashAlg ByteString
m of
            Right Integer
_ -> forall (m :: * -> *) a. Monad m => a -> m a
return ByteString
padding
            Either Error Integer
_       -> m ByteString
findPadding

-- | Calculate hash of message and padding.
-- If the padding is valid, then the result of the hash operation is returned, otherwise an error.
calculateHash :: HashAlgorithm hash
              => ByteString    -- ^ padding
              -> PrivateKey    -- ^ private key
              -> hash          -- ^ hash function
              -> ByteString    -- ^ message to sign
              -> Either Error Integer
calculateHash :: forall hash.
HashAlgorithm hash =>
ByteString
-> PrivateKey -> hash -> ByteString -> Either Error Integer
calculateHash ByteString
padding PrivateKey
pk hash
hashAlg ByteString
m = 
    let p :: Integer
p = PrivateKey -> Integer
private_p PrivateKey
pk
        q :: Integer
q = PrivateKey -> Integer
private_q PrivateKey
pk
        h :: Integer
h = forall ba. ByteArrayAccess ba => ba -> Integer
os2ip forall a b. (a -> b) -> a -> b
$ forall ba alg.
(ByteArrayAccess ba, HashAlgorithm alg) =>
alg -> ba -> Digest alg
hashWith hash
hashAlg forall a b. (a -> b) -> a -> b
$ ByteString -> ByteString -> ByteString
B.append ByteString
padding ByteString
m
     in case (Integer -> Integer -> Maybe Integer
jacobi (Integer
h forall a. Integral a => a -> a -> a
`mod` Integer
p) Integer
p, Integer -> Integer -> Maybe Integer
jacobi (Integer
h forall a. Integral a => a -> a -> a
`mod` Integer
q) Integer
q) of
            (Just Integer
1, Just Integer
1) -> forall a b. b -> Either a b
Right Integer
h
            (Maybe Integer, Maybe Integer)
_                -> forall a b. a -> Either a b
Left Error
InvalidParameters

-- | Verify signature using hash algorithm and public key.
--
-- See <https://en.wikipedia.org/wiki/Rabin_signature_algorithm>.
verify :: HashAlgorithm hash
       => PublicKey     -- ^ private key
       -> hash          -- ^ hash function
       -> ByteString    -- ^ message
       -> Signature     -- ^ signature
       -> Bool
verify :: forall hash.
HashAlgorithm hash =>
PublicKey -> hash -> ByteString -> Signature -> Bool
verify PublicKey
pk hash
hashAlg ByteString
m (Signature (Integer
padding, Integer
s)) =
    let n :: Integer
n  = PublicKey -> Integer
public_n PublicKey
pk
        p :: ByteString
p  = forall ba. ByteArray ba => Integer -> ba
i2osp Integer
padding
        h :: Integer
h  = forall ba. ByteArrayAccess ba => ba -> Integer
os2ip forall a b. (a -> b) -> a -> b
$ forall ba alg.
(ByteArrayAccess ba, HashAlgorithm alg) =>
alg -> ba -> Digest alg
hashWith hash
hashAlg forall a b. (a -> b) -> a -> b
$ ByteString -> ByteString -> ByteString
B.append ByteString
p ByteString
m 
        h' :: Integer
h' = Integer -> Integer -> Integer -> Integer
expSafe Integer
s Integer
2 Integer
n
     in Integer
h' forall a. Eq a => a -> a -> Bool
== Integer
h

-- | Square roots modulo prime p where p is congruent 3 mod 4
-- Value a must be a quadratic residue modulo p (i.e. jacobi symbol (a/n) = 1).
--
-- See algorithm 3.36 in "Handbook of Applied Cryptography" by Alfred J. Menezes et al.
sqroot :: Integer
       -> Integer   -- ^ prime p
       -> (Integer, Integer)
sqroot :: Integer -> Integer -> (Integer, Integer)
sqroot Integer
a Integer
p =
    let r :: Integer
r = Integer -> Integer -> Integer -> Integer
expSafe Integer
a ((Integer
p forall a. Num a => a -> a -> a
+ Integer
1) forall a. Integral a => a -> a -> a
`div` Integer
4) Integer
p
     in (Integer
r, -Integer
r)

-- | Square roots modulo n given its prime factors p and q (both congruent 3 mod 4)
-- Value a must be a quadratic residue of both modulo p and modulo q (i.e. jacobi symbols (a/p) = (a/q) = 1).
-- 
-- See algorithm 3.44 in "Handbook of Applied Cryptography" by Alfred J. Menezes et al.
sqroot' :: Integer 
        -> Integer  -- ^ prime p
        -> Integer  -- ^ prime q
        -> Integer  -- ^ c such that c*p + d*q = 1
        -> Integer  -- ^ d such that c*p + d*q = 1
        -> Integer  -- ^ n = p*q
        -> (Integer, Integer, Integer, Integer)
sqroot' :: Integer
-> Integer
-> Integer
-> Integer
-> Integer
-> Integer
-> (Integer, Integer, Integer, Integer)
sqroot' Integer
a Integer
p Integer
q Integer
c Integer
d Integer
n =
    let (Integer
r, Integer
_) = Integer -> Integer -> (Integer, Integer)
sqroot Integer
a Integer
p
        (Integer
s, Integer
_) = Integer -> Integer -> (Integer, Integer)
sqroot Integer
a Integer
q
        x :: Integer
x      = (Integer
rforall a. Num a => a -> a -> a
*Integer
dforall a. Num a => a -> a -> a
*Integer
q forall a. Num a => a -> a -> a
+ Integer
sforall a. Num a => a -> a -> a
*Integer
cforall a. Num a => a -> a -> a
*Integer
p) forall a. Integral a => a -> a -> a
`mod` Integer
n
        y :: Integer
y      = (Integer
rforall a. Num a => a -> a -> a
*Integer
dforall a. Num a => a -> a -> a
*Integer
q forall a. Num a => a -> a -> a
- Integer
sforall a. Num a => a -> a -> a
*Integer
cforall a. Num a => a -> a -> a
*Integer
p) forall a. Integral a => a -> a -> a
`mod` Integer
n
     in (Integer
x, (-Integer
x) forall a. Integral a => a -> a -> a
`mod` Integer
n, Integer
y, (-Integer
y) forall a. Integral a => a -> a -> a
`mod` Integer
n)