{-# OPTIONS_GHC -fno-warn-orphans #-} module Protocol.Arithmetic ( module Protocol.Arithmetic , Natural ) where import Control.Arrow (first) 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 (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) inF :: forall p i. PrimeField p => Integral i => i -> F p inF i = F (abs (fromIntegral i `mod` fieldCharac @p)) where abs x | x < 0 = x + fieldCharac @p | otherwise = x 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) -- | @('natG' g)@ returns the element of the 'SubGroup' 'g' -- as an 'Natural' within @[0..'fieldCharac'-1]@. natG :: SubGroup q => G q -> Natural natG = 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 -- | @('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 ",") ((\g -> fromString (show (natG g))) <$> gs) in let h = ByteArray.convert (Crypto.hashWith Crypto.SHA256 s) in inE (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) inE :: forall q i. SubGroup q => Integral i => i -> E q inE i = E (F (abs (fromIntegral i `mod` unF (groupOrder @q)))) where abs x | x < 0 = x + unF (groupOrder @q) | otherwise = x natE :: forall q. SubGroup q => E q -> Natural natE = 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 = inE fromEnum = fromIntegral . natE 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 -- * Type 'RandomGen' type RandomGen = Random.RandomGen -- | @('randomR' i)@ returns a random integer in @[0..i-1]@. randomR :: Monad m => 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 => 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 '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