1 {-# LANGUAGE AllowAmbiguousTypes #-}
2 {-# LANGUAGE DeriveAnyClass #-}
3 {-# LANGUAGE DeriveGeneric #-}
4 {-# LANGUAGE DerivingStrategies #-}
5 {-# LANGUAGE OverloadedStrings #-}
6 {-# LANGUAGE InstanceSigs #-}
7 {-# LANGUAGE UndecidableInstances #-} -- for Reifies instances
8 {-# OPTIONS_GHC -fno-warn-orphans #-}
9 -- | Finite Field Cryptography (FFC)
10 -- is a method of implementing discrete logarithm cryptography
11 -- using finite field mathematics.
12 module Voting.Protocol.FFC where
14 import Control.Arrow (first)
15 import Control.DeepSeq (NFData)
16 import Control.Monad (Monad(..), unless)
17 import Data.Aeson (ToJSON(..),FromJSON(..),(.:),(.:?),(.=))
19 import Data.Either (Either(..))
20 import Data.Eq (Eq(..))
21 import Data.Function (($), (.))
22 import Data.Functor ((<$>))
23 import Data.Maybe (Maybe(..), fromMaybe, fromJust)
24 import Data.Monoid (Monoid(..))
25 import Data.Ord (Ord(..))
26 import Data.Proxy (Proxy(..))
27 import Data.Reflection (Reifies(..))
28 import Data.Semigroup (Semigroup(..))
29 import Data.Text (Text)
30 import GHC.Generics (Generic)
31 import GHC.Natural (minusNaturalMaybe)
32 import Numeric.Natural (Natural)
33 import Prelude (Integral(..), fromIntegral)
34 import Text.Read (readMaybe, readEither)
35 import Text.Show (Show(..))
36 import qualified Crypto.KDF.PBKDF2 as Crypto
37 import qualified Data.Aeson as JSON
38 import qualified Data.Aeson.Types as JSON
39 import qualified Data.Char as Char
40 import qualified Data.Text as Text
41 import qualified Data.Text.Encoding as Text
42 import qualified System.Random as Random
44 import Voting.Protocol.Arith
45 import Voting.Protocol.Credential
48 -- | Mutiplicative Sub-Group of a Finite Prime Field.
50 -- NOTE: an 'FFC' term-value is brought into the context of many functions
51 -- through a type-variable @c@ whose 'Reifies' constraint enables to 'reflect'
52 -- that 'FFC' at the term-level (a surprising technique but a very useful one).
53 -- Doing like this is simpler than working in a 'Monad' (like a 'Reader'),
54 -- and enables that 'FFC' term to be used simply in instances' methods
55 -- not supporting an inner 'Monad', like 'parseJSON', 'randomR', 'fromEnum' or 'arbitrary'.
56 -- Aside from that, the sharing of 'FFC' amongst several types
57 -- is encoded at the type-level by including @c@
58 -- as a phantom type of 'F', 'G' and 'E'.
61 , ffc_fieldCharac :: !Natural
62 -- ^ The prime number characteristic of a Finite Prime Field.
64 -- ElGamal's hardness to decrypt requires a large prime number
65 -- to form the 'Multiplicative' subgroup.
66 , ffc_groupGen :: !Natural
67 -- ^ A generator of the 'Multiplicative' subgroup of the Finite Prime Field.
69 -- NOTE: since 'ffc_fieldCharac' is prime,
70 -- the 'Multiplicative' subgroup is cyclic,
71 -- and there are phi('fieldCharac'-1) many choices for the generator of the group,
72 -- where phi is the Euler totient function.
73 , ffc_groupOrder :: !Natural
74 -- ^ The order of the subgroup.
76 -- WARNING: 'ffc_groupOrder' MUST be a prime number dividing @('ffc_fieldCharac'-1)@
77 -- to ensure that ElGamal is secure in terms of the DDH assumption.
78 } deriving (Eq,Show,Generic,NFData)
79 instance ToJSON FFC where
82 (if Text.null ffc_name then [] else ["name" .= ffc_name] ) <>
83 [ "p" .= show ffc_fieldCharac
84 , "g" .= show ffc_groupGen
85 , "q" .= show ffc_groupOrder
89 (if Text.null ffc_name then mempty else "name" .= ffc_name) <>
90 "p" .= show ffc_fieldCharac <>
91 "g" .= show ffc_groupGen <>
92 "q" .= show ffc_groupOrder
93 instance FromJSON FFC where
94 parseJSON = JSON.withObject "FFC" $ \o -> do
95 ffc_name <- fromMaybe "" <$> (o .:? "name")
99 -- TODO: check p is probable prime
100 -- TODO: check q is probable prime
101 ffc_fieldCharac <- case readEither (Text.unpack p) of
102 Left err -> JSON.typeMismatch ("FFC: fieldCharac: "<>err) (JSON.String p)
104 ffc_groupGen <- case readEither (Text.unpack g) of
105 Left err -> JSON.typeMismatch ("FFC: groupGen: "<>err) (JSON.String g)
107 ffc_groupOrder <- case readEither (Text.unpack q) of
108 Left err -> JSON.typeMismatch ("FFC: groupOrder: "<>err) (JSON.String q)
110 unless (nat ffc_groupGen < ffc_fieldCharac) $
111 JSON.typeMismatch "FFC: groupGen is not lower than fieldCharac" (JSON.Object o)
112 unless (ffc_groupOrder < ffc_fieldCharac) $
113 JSON.typeMismatch "FFC: groupOrder is not lower than fieldCharac" (JSON.Object o)
114 unless (nat ffc_groupGen > 1) $
115 JSON.typeMismatch "FFC: groupGen is not greater than 1" (JSON.Object o)
116 unless (fromJust (ffc_fieldCharac`minusNaturalMaybe`one) `rem` ffc_groupOrder == 0) $
117 JSON.typeMismatch "FFC: groupOrder does not divide fieldCharac-1" (JSON.Object o)
119 instance Group FFC where
120 groupGen :: forall c. Reifies c FFC => G FFC c
121 groupGen = G $ F $ ffc_groupGen $ reflect (Proxy::Proxy c)
122 groupOrder :: forall c. Reifies c FFC => Proxy c -> Natural
123 groupOrder c = ffc_groupOrder $ reflect c
125 fieldCharac :: forall c. Reifies c FFC => Natural
126 fieldCharac = ffc_fieldCharac $ reflect (Proxy::Proxy c)
129 -- | Weak parameters for debugging purposes only.
132 { ffc_name = "weakFFC"
133 , ffc_fieldCharac = 263
135 , ffc_groupOrder = 131
138 -- | Parameters used in Belenios.
139 -- A 2048-bit 'fieldCharac' of a Finite Prime Field,
140 -- with a 256-bit 'groupOrder' for a 'Multiplicative' subgroup
141 -- generated by 'groupGen'.
144 { ffc_name = "beleniosFFC"
145 , ffc_fieldCharac = 20694785691422546401013643657505008064922989295751104097100884787057374219242717401922237254497684338129066633138078958404960054389636289796393038773905722803605973749427671376777618898589872735865049081167099310535867780980030790491654063777173764198678527273474476341835600035698305193144284561701911000786737307333564123971732897913240474578834468260652327974647951137672658693582180046317922073668860052627186363386088796882120769432366149491002923444346373222145884100586421050242120365433561201320481118852408731077014151666200162313177169372189248078507711827842317498073276598828825169183103125680162072880719
146 , ffc_groupGen = 2402352677501852209227687703532399932712287657378364916510075318787663274146353219320285676155269678799694668298749389095083896573425601900601068477164491735474137283104610458681314511781646755400527402889846139864532661215055797097162016168270312886432456663834863635782106154918419982534315189740658186868651151358576410138882215396016043228843603930989333662772848406593138406010231675095763777982665103606822406635076697764025346253773085133173495194248967754052573659049492477631475991575198775177711481490920456600205478127054728238140972518639858334115700568353695553423781475582491896050296680037745308460627
147 , ffc_groupOrder = 78571733251071885079927659812671450121821421258408794611510081919805623223441
151 -- | The type of the elements of a Finite Prime Field.
153 -- A field must satisfy the following properties:
155 -- * @(f, ('+'), 'zero')@ forms an abelian group,
156 -- called the 'Additive' group of 'f'.
158 -- * @('NonNull' f, ('*'), 'one')@ forms an abelian group,
159 -- called the 'Multiplicative' group of 'f'.
161 -- * ('*') is associative:
162 -- @(a'*'b)'*'c == a'*'(b'*'c)@ and
163 -- @a'*'(b'*'c) == (a'*'b)'*'c@.
165 -- * ('*') and ('+') are both commutative:
166 -- @a'*'b == b'*'a@ and
169 -- * ('*') and ('+') are both left and right distributive:
170 -- @a'*'(b'+'c) == (a'*'b) '+' (a'*'c)@ and
171 -- @(a'+'b)'*'c == (a'*'c) '+' (b'*'c)@
173 -- The 'Natural' is always within @[0..'fieldCharac'-1]@.
174 newtype F c = F { unF :: Natural }
175 deriving (Eq,Ord,Show)
176 deriving newtype NFData
177 type instance FieldElement FFC = F
178 instance Reifies c FFC => FromJSON (F c) where
179 parseJSON (JSON.String s)
180 | Just (c0,_) <- Text.uncons s
182 , Text.all Char.isDigit s
183 , Just x <- readMaybe (Text.unpack s)
186 parseJSON json = JSON.typeMismatch "FieldElement FFC" json
187 instance Reifies c FFC => FromJSON (G FFC c) where
188 parseJSON (JSON.String s)
189 | Just (c0,_) <- Text.uncons s
191 , Text.all Char.isDigit s
192 , Just x <- readMaybe (Text.unpack s)
195 , r ^ E (groupOrder @FFC (Proxy @c)) == one
197 parseJSON json = JSON.typeMismatch "GroupElement" json
198 instance ToJSON (F c) where
199 toJSON (F x) = JSON.toJSON (show x)
200 instance Reifies c FFC => FromNatural (F c) where
201 fromNatural i = F $ abs $ i `mod` fieldCharac @c
203 abs x | x < 0 = x + fieldCharac @c
205 instance ToNatural (F c) where
207 instance Reifies c FFC => Additive (F c) where
209 F x + F y = F $ (x + y) `mod` fieldCharac @c
210 instance Reifies c FFC => Negable (F c) where
213 | otherwise = F $ fromJust $ nat (fieldCharac @c)`minusNaturalMaybe`x
214 instance Reifies c FFC => Multiplicative (F c) where
216 F x * F y = F $ (x * y) `mod` fieldCharac @c
217 instance Reifies c FFC => Random.Random (F c) where
218 randomR (F lo, F hi) =
219 first (F . fromIntegral) .
222 , toInteger hi`min`(toInteger (fieldCharac @c) - 1) )
224 first (F . fromIntegral) .
225 Random.randomR (0, toInteger (fieldCharac @c) - 1)
229 instance Key FFC where
231 cryptoName = ffc_name
232 randomSecretKey = random
233 credentialSecretKey (UUID uuid) (Credential cred) =
234 fromNatural $ decodeBigEndian $
235 Crypto.fastPBKDF2_SHA256
237 { Crypto.iterCounts = 1000
238 , Crypto.outputLength = 32 -- bytes, ie. 256 bits
240 (Text.encodeUtf8 cred)
241 (Text.encodeUtf8 uuid)
242 publicKey = (groupGen @FFC ^)