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path: root/src/Voting/Protocol/Arithmetic.hs
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{-# OPTIONS_GHC -fno-warn-orphans #-}
module Voting.Protocol.Arithmetic
 ( module Voting.Protocol.Arithmetic
 , Natural
 , Random.RandomGen
 ) where

import Control.Arrow (first)
import Control.DeepSeq (NFData)
import Control.Monad (Monad(..))
import Data.Bits
import Data.Bool
import Data.Eq (Eq(..))
import Data.Foldable (Foldable, foldl')
import Data.Function (($), (.))
import Data.Functor ((<$>))
import Data.Int (Int)
import Data.Maybe (Maybe(..))
import Data.Ord (Ord(..))
import Data.Semigroup (Semigroup(..))
import Data.String (String, IsString(..))
import Numeric.Natural (Natural)
import Prelude (Integer, Integral(..), fromIntegral, Enum(..))
import Text.Show (Show(..))
import qualified Control.Monad.Trans.State.Strict as S
import qualified Crypto.Hash as Crypto
import qualified Data.ByteArray as ByteArray
import qualified Data.ByteString as BS
import qualified Data.List as List
import qualified Prelude as Num
import qualified System.Random as Random

-- * Type 'F'
-- | The type of the elements of a 'PrimeField'.
--
-- A field must satisfy the following properties:
--
-- * @(f, ('+'), 'zero')@ forms an abelian group,
--   called the 'Additive' group of 'f'.
--
-- * @('NonNull' f, ('*'), 'one')@ forms an abelian group,
--   called the 'Multiplicative' group of 'f'.
--
-- * ('*') is associative:
--   @(a'*'b)'*'c == a'*'(b'*'c)@ and
--   @a'*'(b'*'c) == (a'*'b)'*'c@.
--
-- * ('*') and ('+') are both commutative:
--   @a'*'b == b'*'a@ and
--   @a'+'b == b'+'a@
--
-- * ('*') and ('+') are both left and right distributive:
--   @a'*'(b'+'c) == (a'*'b) '+' (a'*'c)@ and
--   @(a'+'b)'*'c == (a'*'c) '+' (b'*'c)@
--
-- The 'Natural' is always within @[0..'fieldCharac'-1]@.
newtype F p = F { unF :: Natural }
 deriving (Eq,Ord,Show,NFData)

instance PrimeField p => FromNatural (F p) where
	fromNatural i = F (abs (i `mod` fieldCharac @p))
		where abs x | x < 0 = x + fieldCharac @p
		            | otherwise = x
instance ToNatural (F p) where
	nat = unF

instance PrimeField p => Additive (F p) where
	zero = F 0
	F x + F y = F ((x + y) `mod` fieldCharac @p)
instance PrimeField p => Negable (F p) where
	neg (F x) | x == 0 = zero
	          | otherwise = F (fromIntegral (Num.negate (toInteger x) + toInteger (fieldCharac @p)))
instance PrimeField p => Multiplicative (F p) where
	one = F 1
	-- | Because 'fieldCharac' is prime,
	-- all elements of the field are invertible modulo 'fieldCharac'.
	F x * F y = F ((x * y) `mod` fieldCharac @p)
instance PrimeField p => Random.Random (F p) where
	randomR (F lo, F hi) =
		first (F . fromIntegral) .
		Random.randomR
		 ( 0`max`toInteger lo
		 , toInteger hi`min`(toInteger (fieldCharac @p) - 1))
	random = first (F . fromIntegral) . Random.randomR (0, toInteger (fieldCharac @p) - 1)

-- ** Class 'PrimeField'
-- | Parameter for a prime field.
class PrimeField p where
	-- | The prime number characteristic of a 'PrimeField'.
	--
	-- ElGamal's hardness to decrypt requires a large prime number
	-- to form the 'Multiplicative' 'SubGroup'.
	fieldCharac :: Natural

-- ** Class 'Additive'
class Additive a where
	zero :: a
	(+) :: a -> a -> a; infixl 6 +
	sum :: Foldable f => f a -> a
	sum = foldl' (+) zero
instance Additive Natural where
	zero = 0
	(+)  = (Num.+)
instance Additive Integer where
	zero = 0
	(+)  = (Num.+)
instance Additive Int where
	zero = 0
	(+)  = (Num.+)

-- *** Class 'Negable'
class Additive a => Negable a where
	neg :: a -> a
	(-) :: a -> a -> a; infixl 6 -
	x-y = x + neg y
instance Negable Integer where
	neg  = Num.negate
instance Negable Int where
	neg  = Num.negate

-- ** Class 'Multiplicative'
class Multiplicative a where
	one :: a
	(*) :: a -> a -> a; infixl 7 *
instance Multiplicative Natural where
	one = 1
	(*) = (Num.*)
instance Multiplicative Integer where
	one = 1
	(*) = (Num.*)
instance Multiplicative Int where
	one = 1
	(*) = (Num.*)

-- ** Class 'Invertible'
class Multiplicative a => Invertible a where
	inv :: a -> a
	(/) :: a -> a -> a; infixl 7 /
	x/y = x * inv y

-- * Type 'G'
-- | The type of the elements of a 'Multiplicative' 'SubGroup' of a 'PrimeField'.
newtype G q = G { unG :: F (P q) }
 deriving (Eq,Ord,Show,NFData)

instance PrimeField (P q) => FromNatural (G q) where
	fromNatural = G . fromNatural
instance ToNatural (G q) where
	nat = unF . unG

instance (SubGroup q, Multiplicative (F (P q))) => Multiplicative (G q) where
	one = G one
	G x * G y = G (x * y)
instance (SubGroup q, Multiplicative (F (P q))) => Invertible (G q) where
	-- | NOTE: add 'groupOrder' so the exponent given to (^) is positive.
	inv = (^ E (neg one + groupOrder @q))

-- ** Class 'SubGroup'
-- | A 'SubGroup' of a 'Multiplicative' group of a 'PrimeField'.
-- Used for signing (Schnorr) and encrypting (ElGamal).
class
 ( PrimeField (P q)
 , Multiplicative (F (P q))
 ) => SubGroup q where
	-- | Setting 'q' determines 'p', equals to @'P' q@.
	type P q :: *
	-- | A generator of the 'SubGroup'.
	-- NOTE: since @F p@ is a 'PrimeField',
	-- the 'Multiplicative' 'SubGroup' is cyclic,
	-- and there are phi('fieldCharac'-1) many choices for the generator of the group,
	-- where phi is the Euler totient function.
	groupGen :: G q
	-- | The order of the 'SubGroup'.
	--
	-- WARNING: 'groupOrder' MUST be a prime number dividing @('fieldCharac'-1)@
	-- to ensure that ElGamal is secure in terms of the DDH assumption.
	groupOrder :: F (P q)
	
	-- | 'groupGenInverses' returns the infinite list
	-- of 'inv'erse powers of 'groupGen':
	-- @['groupGen' '^' 'neg' i | i <- [0..]]@,
	-- but by computing each value from the previous one.
	--
	-- NOTE: 'groupGenInverses' is in the 'SubGroup' class in order to keep
	-- computed terms in memory across calls to 'groupGenInverses'.
	--
	-- Used by 'intervalDisjunctions'.
	groupGenInverses :: [G q]
	groupGenInverses = go one
		where
		go g = g : go (g * invGen)
		invGen = inv groupGen

groupGenPowers :: SubGroup q => [G q]
groupGenPowers = go one
	where go g = g : go (g * groupGen)

-- | @('hash' bs gs)@ returns as a number in 'E'
-- the SHA256 of the given 'BS.ByteString' 'bs'
-- prefixing the decimal representation of given 'SubGroup' elements 'gs',
-- with a comma (",") intercalated between them.
--
-- NOTE: to avoid any collision when the 'hash' function is used in different contexts,
-- a message 'gs' is actually prefixed by a 'bs' indicating the context.
--
-- Used by 'proveEncryption' and 'verifyEncryption',
-- where the 'bs' usually contains the 'statement' to be proven,
-- and the 'gs' contains the 'commitments'.
hash :: SubGroup q => BS.ByteString -> [G q] -> E q
hash bs gs =
	let s = bs <> BS.intercalate (fromString ",") (bytesNat <$> gs) in
	let h = ByteArray.convert (Crypto.hashWith Crypto.SHA256 s) in
	fromNatural (BS.foldl' (\acc b -> acc`shiftL`3 + fromIntegral b) (0::Natural) h)

-- * Type 'E'
-- | An exponent of a (necessarily cyclic) 'SubGroup' of a 'PrimeField'.
-- The value is always in @[0..'groupOrder'-1]@.
newtype E q = E { unE :: F (P q) }
 deriving (Eq,Ord,Show,NFData)

instance SubGroup q => FromNatural (E q) where
	fromNatural i = E (F (abs (i `mod` unF (groupOrder @q))))
		where abs x | x < 0 = x + unF (groupOrder @q)
		            | otherwise = x
instance ToNatural (E q) where
	nat = unF . unE

instance (SubGroup q, Additive (F (P q))) => Additive (E q) where
	zero = E zero
	E (F x) + E (F y) = E (F ((x + y) `mod` unF (groupOrder @q)))
instance (SubGroup q, Negable (F (P q))) => Negable (E q) where
	neg (E (F x)) | x == 0 = zero
	              | otherwise = E (F (fromIntegral ( neg (toInteger x)
	                                               + toInteger (unF (groupOrder @q)) )))
instance (SubGroup q, Multiplicative (F (P q))) => Multiplicative (E q) where
	one = E one
	E (F x) * E (F y) = E (F ((x * y) `mod` unF (groupOrder @q)))
instance SubGroup q => Random.Random (E q) where
	randomR (E (F lo), E (F hi)) =
		first (E . F . fromIntegral) .
		Random.randomR
		 ( 0`max`toInteger lo
		 , toInteger hi`min`(toInteger (unF (groupOrder @q)) - 1) )
	random =
		first (E . F . fromIntegral) .
		Random.randomR (0, toInteger (unF (groupOrder @q)) - 1)
instance SubGroup q => Enum (E q) where
	toEnum = fromNatural . fromIntegral
	fromEnum = fromIntegral . nat
	enumFromTo lo hi = List.unfoldr
	 (\i -> if i<=hi then Just (i, i+one) else Nothing) lo

infixr 8 ^
-- | @(b '^' e)@ returns the modular exponentiation of base 'b' by exponent 'e'.
(^) :: SubGroup q => G q -> E q -> G q
(^) b (E (F e))
 | e == zero = one
 | otherwise = t * (b*b) ^ E (F (e`shiftR`1))
	where
	t | testBit e 0 = b
		| otherwise   = one

-- | @('randomR' i)@ returns a random integer in @[0..i-1]@.
randomR ::
 Monad m =>
 Random.RandomGen r =>
 Random.Random i =>
 Negable i =>
 Multiplicative i =>
 i -> S.StateT r m i
randomR i = S.StateT $ return . Random.randomR (zero, i-one)

-- | @('random')@ returns a random integer
-- in the range determined by its type.
random ::
 Monad m =>
 Random.RandomGen r =>
 Random.Random i =>
 Negable i =>
 Multiplicative i =>
 S.StateT r m i
random = S.StateT $ return . Random.random

instance Random.Random Natural where
	randomR (mini,maxi) =
		first (fromIntegral::Integer -> Natural) .
		Random.randomR (fromIntegral mini, fromIntegral maxi)
	random = first (fromIntegral::Integer -> Natural) . Random.random

-- * Groups

-- * Type 'Params'
class SubGroup q => Params q where
	paramsName :: String
instance Params WeakParams where
	paramsName = "WeakParams"
instance Params BeleniosParams where
	paramsName = "BeleniosParams"

-- ** Type 'WeakParams'
-- | Weak parameters for debugging purposes only.
data WeakParams
instance PrimeField WeakParams where
	fieldCharac = 263
instance SubGroup WeakParams where
	type P WeakParams = WeakParams
	groupGen = G (F 2)
	groupOrder = F 131

-- ** Type 'BeleniosParams'
-- | Parameters used in Belenios.
-- A 2048-bit 'fieldCharac' of a 'PrimeField',
-- with a 256-bit 'groupOrder' for a 'Multiplicative' 'SubGroup'
-- generated by 'groupGen'.
data BeleniosParams
instance PrimeField BeleniosParams where
	fieldCharac = 20694785691422546401013643657505008064922989295751104097100884787057374219242717401922237254497684338129066633138078958404960054389636289796393038773905722803605973749427671376777618898589872735865049081167099310535867780980030790491654063777173764198678527273474476341835600035698305193144284561701911000786737307333564123971732897913240474578834468260652327974647951137672658693582180046317922073668860052627186363386088796882120769432366149491002923444346373222145884100586421050242120365433561201320481118852408731077014151666200162313177169372189248078507711827842317498073276598828825169183103125680162072880719
instance SubGroup BeleniosParams where
	type P BeleniosParams = BeleniosParams
	groupGen = G (F 2402352677501852209227687703532399932712287657378364916510075318787663274146353219320285676155269678799694668298749389095083896573425601900601068477164491735474137283104610458681314511781646755400527402889846139864532661215055797097162016168270312886432456663834863635782106154918419982534315189740658186868651151358576410138882215396016043228843603930989333662772848406593138406010231675095763777982665103606822406635076697764025346253773085133173495194248967754052573659049492477631475991575198775177711481490920456600205478127054728238140972518639858334115700568353695553423781475582491896050296680037745308460627)
	groupOrder = F 78571733251071885079927659812671450121821421258408794611510081919805623223441

-- * Conversions

-- ** Class 'FromNatural'
class FromNatural a where
	fromNatural :: Natural -> a

-- ** Class 'ToNatural'
class ToNatural a where
	nat :: a -> Natural

-- | @('bytesNat' x)@ returns the serialization of 'x'.
bytesNat :: ToNatural n => n -> BS.ByteString
bytesNat = fromString . show . nat