Protocol.Arithmetic

Type F

newtype F p

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].

Constructors

F

Fields

• unF :: Natural

Instances

Eq (F p)

Defined in Protocol.Arithmetic

Methods

(==) :: F p -> F p -> Bool

(/=) :: F p -> F p -> Bool

Ord (F p)

Defined in Protocol.Arithmetic

Methods

compare :: F p -> F p -> Ordering

(<) :: F p -> F p -> Bool

(<=) :: F p -> F p -> Bool

(>) :: F p -> F p -> Bool

(>=) :: F p -> F p -> Bool

max :: F p -> F p -> F p

min :: F p -> F p -> F p

Show (F p)

Defined in Protocol.Arithmetic

Methods

showsPrec :: Int -> F p -> ShowS

show :: F p -> String

showList :: [F p] -> ShowS

PrimeField p => Random (F p)

Defined in Protocol.Arithmetic

Methods

randomR :: RandomGen g => (F p, F p) -> g -> (F p, g)

random :: RandomGen g => g -> (F p, g)

randomRs :: RandomGen g => (F p, F p) -> g -> [F p]

randoms :: RandomGen g => g -> [F p]

randomRIO :: (F p, F p) -> IO (F p)

randomIO :: IO (F p)

PrimeField p => Multiplicative (F p)

Defined in Protocol.Arithmetic

Methods

one :: F p

(*) :: F p -> F p -> F p

PrimeField p => Negable (F p)

Defined in Protocol.Arithmetic

Methods

neg :: F p -> F p

(-) :: F p -> F p -> F p

PrimeField p => Additive (F p)

Defined in Protocol.Arithmetic

Methods

zero :: F p

(+) :: F p -> F p -> F p

sum :: Foldable f => f (F p) -> F p

inF :: forall p i. PrimeField p => Integral i => i -> F p

Class PrimeField

class PrimeField p where

Parameter for a prime field.

Methods

fieldCharac :: Natural

The prime number characteristic of a PrimeField.

ElGamal's hardness to decrypt requires a large prime number to form the Multiplicative SubGroup.

Instances

PrimeField BeleniosParams

Defined in Protocol.Arithmetic

Methods

fieldCharac :: Natural

PrimeField WeakParams

Defined in Protocol.Arithmetic

Methods

fieldCharac :: Natural

Class Additive

class Additive a where

Minimal complete definition

zero, (+)

Methods

zero :: a

(+) :: a -> a -> a infixl 6

sum :: Foldable f => f a -> a

Instances

Additive Int

Defined in Protocol.Arithmetic

Methods

zero :: Int

(+) :: Int -> Int -> Int

sum :: Foldable f => f Int -> Int

Additive Integer

Defined in Protocol.Arithmetic

Methods

zero :: Integer

(+) :: Integer -> Integer -> Integer

sum :: Foldable f => f Integer -> Integer

Additive Natural

Defined in Protocol.Arithmetic

Methods

zero :: Natural

(+) :: Natural -> Natural -> Natural

sum :: Foldable f => f Natural -> Natural

(SubGroup q, Additive (F (P q))) => Additive (E q)

Defined in Protocol.Arithmetic

Methods

zero :: E q

(+) :: E q -> E q -> E q

sum :: Foldable f => f (E q) -> E q

PrimeField p => Additive (F p)

Defined in Protocol.Arithmetic

Methods

zero :: F p

(+) :: F p -> F p -> F p

sum :: Foldable f => f (F p) -> F p

SubGroup q => Additive (Encryption q)

Additive homomorphism. Using the fact that: groupGen ^x * groupGen ^y == groupGen ^(x+y).

Defined in Protocol.Election

Methods

zero :: Encryption q

(+) :: Encryption q -> Encryption q -> Encryption q

sum :: Foldable f => f (Encryption q) -> Encryption q

Class Negable

class Additive a => Negable a where

Minimal complete definition

neg

Methods

neg :: a -> a

(-) :: a -> a -> a infixl 6

Instances

Negable Int

Defined in Protocol.Arithmetic

Methods

neg :: Int -> Int

(-) :: Int -> Int -> Int

Negable Integer

Defined in Protocol.Arithmetic

Methods

neg :: Integer -> Integer

(-) :: Integer -> Integer -> Integer

(SubGroup q, Negable (F (P q))) => Negable (E q)

Defined in Protocol.Arithmetic

Methods

neg :: E q -> E q

(-) :: E q -> E q -> E q

PrimeField p => Negable (F p)

Defined in Protocol.Arithmetic

Methods

neg :: F p -> F p

(-) :: F p -> F p -> F p

Class Multiplicative

class Multiplicative a where

Methods

one :: a

(*) :: a -> a -> a infixl 7

Instances

Multiplicative Int

Defined in Protocol.Arithmetic

Methods

one :: Int

(*) :: Int -> Int -> Int

Multiplicative Integer

Defined in Protocol.Arithmetic

Methods

one :: Integer

(*) :: Integer -> Integer -> Integer

Multiplicative Natural

Defined in Protocol.Arithmetic

Methods

one :: Natural

(*) :: Natural -> Natural -> Natural

(SubGroup q, Multiplicative (F (P q))) => Multiplicative (E q)

Defined in Protocol.Arithmetic

Methods

one :: E q

(*) :: E q -> E q -> E q

(SubGroup q, Multiplicative (F (P q))) => Multiplicative (G q)

Defined in Protocol.Arithmetic

Methods

one :: G q

(*) :: G q -> G q -> G q

PrimeField p => Multiplicative (F p)

Defined in Protocol.Arithmetic

Methods

one :: F p

(*) :: F p -> F p -> F p

Class Invertible

class Multiplicative a => Invertible a where

Minimal complete definition

inv

Methods

inv :: a -> a

(/) :: a -> a -> a infixl 7

Instances

(SubGroup q, Multiplicative (F (P q))) => Invertible (G q)

Defined in Protocol.Arithmetic

Methods

inv :: G q -> G q

(/) :: G q -> G q -> G q

Type G

newtype G q

The type of the elements of a Multiplicative SubGroup of a PrimeField.

Constructors

G

Fields

• unG :: F (P q)

Instances

Eq (G q)

Defined in Protocol.Arithmetic

Methods

(==) :: G q -> G q -> Bool

(/=) :: G q -> G q -> Bool

Ord (G q)

Defined in Protocol.Arithmetic

Methods

compare :: G q -> G q -> Ordering

(<) :: G q -> G q -> Bool

(<=) :: G q -> G q -> Bool

(>) :: G q -> G q -> Bool

(>=) :: G q -> G q -> Bool

max :: G q -> G q -> G q

min :: G q -> G q -> G q

Show (G q)

Defined in Protocol.Arithmetic

Methods

showsPrec :: Int -> G q -> ShowS

show :: G q -> String

showList :: [G q] -> ShowS

(SubGroup q, Multiplicative (F (P q))) => Invertible (G q)

Defined in Protocol.Arithmetic

Methods

inv :: G q -> G q

(/) :: G q -> G q -> G q

(SubGroup q, Multiplicative (F (P q))) => Multiplicative (G q)

Defined in Protocol.Arithmetic

Methods

one :: G q

(*) :: G q -> G q -> G q

natG :: SubGroup q => G q -> Natural

(natG g) returns the element of the SubGroup g as an Natural within [0..fieldCharac-1].

Class SubGroup

class (PrimeField (P q), Multiplicative (F (P q))) => SubGroup q where

A SubGroup of a PrimeField. Used for signing (Schnorr) and encrypting (ElGamal).

Minimal complete definition

groupGen, groupOrder

Associated Types

type P q :: *

Setting q determines p, equals to P q.

Methods

groupGen :: G 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.

groupOrder :: F (P q)

The order of the SubGroup.

WARNING: groupOrder MUST be a prime number dividing (fieldCharac-1) to ensure that ensures that ElGamal is secure in terms of the DDH assumption.

groupGenInverses :: [G q]

groupGenInverses returns the infinite list of inverse 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 accross calls to groupGenInverses.

Used by validableEncryption.

Instances

SubGroup BeleniosParams

Defined in Protocol.Arithmetic

Associated Types

type P BeleniosParams :: Type

Methods

groupGen :: G BeleniosParams

groupOrder :: F (P BeleniosParams)

groupGenInverses :: [G BeleniosParams]

SubGroup WeakParams

Defined in Protocol.Arithmetic

Associated Types

type P WeakParams :: Type

Methods

groupGen :: G WeakParams

groupOrder :: F (P WeakParams)

groupGenInverses :: [G WeakParams]

hash :: SubGroup q => ByteString -> [G q] -> E q

(hash prefix gs) returns as a number in (F p) the SHA256 of the given prefix prefixing the decimal representation of given SubGroup elements gs, each one postfixed with a comma (",").

Used by proveEncryption and validateEncryption, where the prefix contains the statement to be proven, and the gs contains the commitments.

Type E

newtype E q

An exponent of a (necessarily cyclic) SubGroup of a PrimeField. The value is always in [0..groupOrder-1].

Constructors

E

Fields

• unE :: F (P q)

Instances

SubGroup q => Enum (E q)

Defined in Protocol.Arithmetic

Methods

succ :: E q -> E q

pred :: E q -> E q

toEnum :: Int -> E q

fromEnum :: E q -> Int

enumFrom :: E q -> [E q]

enumFromThen :: E q -> E q -> [E q]

enumFromTo :: E q -> E q -> [E q]

enumFromThenTo :: E q -> E q -> E q -> [E q]

Eq (E q)

Defined in Protocol.Arithmetic

Methods

(==) :: E q -> E q -> Bool

(/=) :: E q -> E q -> Bool

Ord (E q)

Defined in Protocol.Arithmetic

Methods

compare :: E q -> E q -> Ordering

(<) :: E q -> E q -> Bool

(<=) :: E q -> E q -> Bool

(>) :: E q -> E q -> Bool

(>=) :: E q -> E q -> Bool

max :: E q -> E q -> E q

min :: E q -> E q -> E q

Show (E q)

Defined in Protocol.Arithmetic

Methods

showsPrec :: Int -> E q -> ShowS

show :: E q -> String

showList :: [E q] -> ShowS

SubGroup q => Random (E q)

Defined in Protocol.Arithmetic

Methods

randomR :: RandomGen g => (E q, E q) -> g -> (E q, g)

random :: RandomGen g => g -> (E q, g)

randomRs :: RandomGen g => (E q, E q) -> g -> [E q]

randoms :: RandomGen g => g -> [E q]

randomRIO :: (E q, E q) -> IO (E q)

randomIO :: IO (E q)

(SubGroup q, Multiplicative (F (P q))) => Multiplicative (E q)

Defined in Protocol.Arithmetic

Methods

one :: E q

(*) :: E q -> E q -> E q

(SubGroup q, Negable (F (P q))) => Negable (E q)

Defined in Protocol.Arithmetic

Methods

neg :: E q -> E q

(-) :: E q -> E q -> E q

(SubGroup q, Additive (F (P q))) => Additive (E q)

Defined in Protocol.Arithmetic

Methods

zero :: E q

(+) :: E q -> E q -> E q

sum :: Foldable f => f (E q) -> E q

inE :: forall q i. SubGroup q => Integral i => i -> E q

natE :: forall q. SubGroup q => E q -> Natural

(^) :: SubGroup q => G q -> E q -> G q infixr 8

(b ^ e) returns the modular exponentiation of base b by exponent e.

Type RandomGen

type RandomGen = RandomGen

randomR :: Monad m => RandomGen r => Random i => Negable i => Multiplicative i => i -> StateT r m i

(randomR i) returns a random integer in [0..i-1].

random :: Monad m => RandomGen r => Random i => Negable i => Multiplicative i => StateT r m i

(random) returns a random integer in the range determined by its type.

Groups

Type WeakParams

data WeakParams

Weak parameters for debugging purposes only.

Instances

SubGroup WeakParams

Defined in Protocol.Arithmetic

Associated Types

type P WeakParams :: Type

Methods

groupGen :: G WeakParams

groupOrder :: F (P WeakParams)

groupGenInverses :: [G WeakParams]

PrimeField WeakParams

Defined in Protocol.Arithmetic

Methods

fieldCharac :: Natural

type P WeakParams

Defined in Protocol.Arithmetic

type P WeakParams = WeakParams

Type BeleniosParams

data BeleniosParams

Parameters used in Belenios. A 2048-bit fieldCharac of a PrimeField, with a 256-bit groupOrder for a Multiplicative SubGroup generated by groupGen.

Instances

SubGroup BeleniosParams

Defined in Protocol.Arithmetic

Associated Types

type P BeleniosParams :: Type

Methods

groupGen :: G BeleniosParams

groupOrder :: F (P BeleniosParams)

groupGenInverses :: [G BeleniosParams]

PrimeField BeleniosParams

Defined in Protocol.Arithmetic

Methods

fieldCharac :: Natural

type P BeleniosParams

Defined in Protocol.Arithmetic

type P BeleniosParams = BeleniosParams

Orphan instances

Random Natural

Methods

randomR :: RandomGen g => (Natural, Natural) -> g -> (Natural, g)

random :: RandomGen g => g -> (Natural, g)

randomRs :: RandomGen g => (Natural, Natural) -> g -> [Natural]

randoms :: RandomGen g => g -> [Natural]

randomRIO :: (Natural, Natural) -> IO Natural

randomIO :: IO Natural