Crypto.Saltine.Core.ScalarMult

Scalar multiplication: Crypto.Saltine.Core.ScalarMult

The mult function multiplies a group element by an integer of length multScalar. It returns the resulting group element of length mult. The multBase function multiplies a standard group element by an integer of length multScalar. It returns the resulting group element of length mult.

The correspondence between strings and group elements depends on the primitive implemented by mult. The correspondence is not necessarily injective in either direction, but it is compatible with scalar multiplication in the group. The correspondence does not necessarily include all group elements, but it does include all strings; i.e., every string represents at least one group element.

The correspondence between strings and integers also depends on the primitive implemented by mult. Every string represents at least one integer.

mult is designed to be strong as a component of various well-known "hashed Diffie–Hellman" applications. In particular, it is designed to make the "computational Diffie–Hellman" problem (CDH) difficult with respect to the standard base. mult is also designed to make CDH difficult with respect to other nontrivial bases. In particular, if a represented group element has small order, then it is annihilated by all represented scalars. This feature allows protocols to avoid validating membership in the subgroup generated by the standard base.

NaCl does not make any promises regarding the "decisional Diffie–Hellman" problem (DDH), the "static Diffie–Hellman" problem (SDH), etc. Users are responsible for hashing group elements.

mult is the function crypto_scalarmult_curve25519 specified in "Cryptography in NaCl", Sections 2, 3, and 4 (http://nacl.cr.yp.to/valid.html). This function is conjectured to be strong. For background see Bernstein, "Curve25519: new Diffie-Hellman speed records," Lecture Notes in Computer Science 3958 (2006), 207–228, http://cr.yp.to/papers.html#curve25519.

This is version 2010.08.30 of the scalarmult.html web page.

Documentation

data Scalar

A scalar integer.

Instances

Data Scalar

Defined in Crypto.Saltine.Internal.ScalarMult

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Scalar -> c Scalar

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c Scalar

toConstr :: Scalar -> Constr

dataTypeOf :: Scalar -> DataType

dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c Scalar)

dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c Scalar)

gmapT :: (forall b. Data b => b -> b) -> Scalar -> Scalar

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Scalar -> r

gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Scalar -> r

gmapQ :: (forall d. Data d => d -> u) -> Scalar -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> Scalar -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> Scalar -> m Scalar

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Scalar -> m Scalar

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Scalar -> m Scalar

Generic Scalar

Defined in Crypto.Saltine.Internal.ScalarMult

Associated Types

type Rep Scalar :: Type -> Type

Methods

from :: Scalar -> Rep Scalar x

to :: Rep Scalar x -> Scalar

Show Scalar

Defined in Crypto.Saltine.Internal.ScalarMult

Methods

showsPrec :: Int -> Scalar -> ShowS

show :: Scalar -> String

showList :: [Scalar] -> ShowS

NFData Scalar

Defined in Crypto.Saltine.Internal.ScalarMult

Methods

rnf :: Scalar -> ()

Eq Scalar

Defined in Crypto.Saltine.Internal.ScalarMult

Methods

(==) :: Scalar -> Scalar -> Bool

(/=) :: Scalar -> Scalar -> Bool

Ord Scalar

Defined in Crypto.Saltine.Internal.ScalarMult

Methods

compare :: Scalar -> Scalar -> Ordering

(<) :: Scalar -> Scalar -> Bool

(<=) :: Scalar -> Scalar -> Bool

(>) :: Scalar -> Scalar -> Bool

(>=) :: Scalar -> Scalar -> Bool

max :: Scalar -> Scalar -> Scalar

min :: Scalar -> Scalar -> Scalar

Hashable Scalar

Defined in Crypto.Saltine.Internal.ScalarMult

Methods

hashWithSalt :: Int -> Scalar -> Int

hash :: Scalar -> Int

IsEncoding Scalar

Defined in Crypto.Saltine.Internal.ScalarMult

Methods

encode :: Scalar -> ByteString

decode :: ByteString -> Maybe Scalar

encoded :: (Choice p, Applicative f) => p Scalar (f Scalar) -> p ByteString (f ByteString)

type Rep Scalar

Defined in Crypto.Saltine.Internal.ScalarMult

type Rep Scalar = D1 ('MetaData "Scalar" "Crypto.Saltine.Internal.ScalarMult" "saltine-0.2.1.0-inplace" 'True) (C1 ('MetaCons "Sc" 'PrefixI 'True) (S1 ('MetaSel ('Just "unSc") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 ByteString)))

data GroupElement

A group element.

Instances

Data GroupElement

Defined in Crypto.Saltine.Internal.ScalarMult

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> GroupElement -> c GroupElement

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c GroupElement

toConstr :: GroupElement -> Constr

dataTypeOf :: GroupElement -> DataType

dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c GroupElement)

dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c GroupElement)

gmapT :: (forall b. Data b => b -> b) -> GroupElement -> GroupElement

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> GroupElement -> r

gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> GroupElement -> r

gmapQ :: (forall d. Data d => d -> u) -> GroupElement -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> GroupElement -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> GroupElement -> m GroupElement

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> GroupElement -> m GroupElement

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> GroupElement -> m GroupElement

Generic GroupElement

Defined in Crypto.Saltine.Internal.ScalarMult

Associated Types

type Rep GroupElement :: Type -> Type

Methods

from :: GroupElement -> Rep GroupElement x

to :: Rep GroupElement x -> GroupElement

Show GroupElement

Defined in Crypto.Saltine.Internal.ScalarMult

Methods

showsPrec :: Int -> GroupElement -> ShowS

show :: GroupElement -> String

showList :: [GroupElement] -> ShowS

NFData GroupElement

Defined in Crypto.Saltine.Internal.ScalarMult

Methods

rnf :: GroupElement -> ()

Eq GroupElement

Defined in Crypto.Saltine.Internal.ScalarMult

Methods

(==) :: GroupElement -> GroupElement -> Bool

(/=) :: GroupElement -> GroupElement -> Bool

Ord GroupElement

Defined in Crypto.Saltine.Internal.ScalarMult

Methods

compare :: GroupElement -> GroupElement -> Ordering

(<) :: GroupElement -> GroupElement -> Bool

(<=) :: GroupElement -> GroupElement -> Bool

(>) :: GroupElement -> GroupElement -> Bool

(>=) :: GroupElement -> GroupElement -> Bool

max :: GroupElement -> GroupElement -> GroupElement

min :: GroupElement -> GroupElement -> GroupElement

Hashable GroupElement

Defined in Crypto.Saltine.Internal.ScalarMult

Methods

hashWithSalt :: Int -> GroupElement -> Int

hash :: GroupElement -> Int

IsEncoding GroupElement

Defined in Crypto.Saltine.Internal.ScalarMult

Methods

encode :: GroupElement -> ByteString

decode :: ByteString -> Maybe GroupElement

encoded :: (Choice p, Applicative f) => p GroupElement (f GroupElement) -> p ByteString (f ByteString)

type Rep GroupElement

Defined in Crypto.Saltine.Internal.ScalarMult

type Rep GroupElement = D1 ('MetaData "GroupElement" "Crypto.Saltine.Internal.ScalarMult" "saltine-0.2.1.0-inplace" 'True) (C1 ('MetaCons "GE" 'PrefixI 'True) (S1 ('MetaSel ('Just "unGE") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 ByteString)))

mult :: Scalar -> GroupElement -> GroupElement

multBase :: Scalar -> GroupElement