-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/


-- | Natural number arithmetic
--   
--   This package implements a library of natural number arithmetic
--   functions, including Montgomery multiplication and continued
--   fractions.
@package arithmetic
@version 1.1


module Arithmetic.Smooth
factorOut :: Natural -> Natural -> Maybe (Natural, Natural)
factorList :: [Natural] -> Natural -> ([(Natural, Natural)], Natural)
factorBase :: Natural -> Natural -> ([(Natural, Natural)], Natural)
multiplyBase :: ([(Natural, Natural)], Natural) -> Natural
newtype Smooth
Smooth :: ([(Natural, Natural)], Natural) -> Smooth
[unSmooth] :: Smooth -> ([(Natural, Natural)], Natural)
fromNatural :: Natural -> Natural -> Smooth
toNatural :: Smooth -> Natural
factoring :: Smooth -> Maybe [(Natural, Natural)]
next :: Natural -> Natural -> Smooth
instance GHC.Classes.Ord Arithmetic.Smooth.Smooth
instance GHC.Classes.Eq Arithmetic.Smooth.Smooth
instance GHC.Show.Show Arithmetic.Smooth.Smooth


module Arithmetic.Random
randomWidth :: Int -> Random -> Natural
randomOdd :: Int -> Random -> Natural
randomCoprime :: Int -> Random -> (Natural, Natural)
uniformInteger :: Integer -> Random -> Integer
randomCoprimeInteger :: Int -> Random -> (Integer, Integer)


module Arithmetic.Modular
multiplyExponential :: (a -> a -> a) -> a -> a -> Natural -> a
functionPower :: (a -> a) -> Natural -> a -> a
normalize :: Natural -> Natural -> Natural
add :: Natural -> Natural -> Natural -> Natural
negate :: Natural -> Natural -> Natural
subtract :: Natural -> Natural -> Natural -> Natural
multiply :: Natural -> Natural -> Natural -> Natural
square :: Natural -> Natural -> Natural
exp :: Natural -> Natural -> Natural -> Natural
exp2 :: Natural -> Natural -> Natural -> Natural
invert :: Natural -> Natural -> Maybe Natural
divide :: Natural -> Natural -> Natural -> Maybe Natural


module Arithmetic.Montgomery
data Parameters
Parameters :: Natural -> Natural -> Natural -> Natural -> Natural -> Natural -> Natural -> Parameters
[nParameters] :: Parameters -> Natural
[wParameters] :: Parameters -> Natural
[sParameters] :: Parameters -> Natural
[kParameters] :: Parameters -> Natural
[rParameters] :: Parameters -> Natural
[r2Parameters] :: Parameters -> Natural
[zParameters] :: Parameters -> Natural
data Montgomery
Montgomery :: Parameters -> Natural -> Montgomery
[pMontgomery] :: Montgomery -> Parameters
[nMontgomery] :: Montgomery -> Natural
align :: Natural -> Natural -> Natural
customParameters :: Natural -> Natural -> Parameters
alignedParameters :: Natural -> Natural -> Parameters
standardParameters :: Natural -> Parameters
normalize :: Parameters -> Natural -> Montgomery
normalize1 :: Parameters -> Natural -> Montgomery
reduce :: Parameters -> Natural -> Natural
toNatural :: Montgomery -> Natural
fromNatural :: Parameters -> Natural -> Montgomery
zero :: Parameters -> Montgomery
one :: Parameters -> Montgomery
two :: Parameters -> Montgomery
add :: Montgomery -> Montgomery -> Montgomery
double :: Montgomery -> Montgomery
negate :: Montgomery -> Montgomery
subtract :: Montgomery -> Montgomery -> Montgomery
multiply :: Montgomery -> Montgomery -> Montgomery
square :: Montgomery -> Montgomery
exp :: Montgomery -> Natural -> Montgomery
exp2 :: Montgomery -> Natural -> Montgomery
modexp :: Natural -> Natural -> Natural -> Natural
modexp2 :: Natural -> Natural -> Natural -> Natural
instance GHC.Show.Show Arithmetic.Montgomery.Montgomery
instance GHC.Show.Show Arithmetic.Montgomery.Parameters


module Arithmetic.Prime
factorTwos :: Natural -> (Int, Natural)
millerRabinWitness :: Natural -> Natural -> Bool
millerRabin :: Int -> Natural -> Random -> Bool
isPrime :: Natural -> Random -> Bool
previousPrime :: Natural -> Random -> Natural
randomPrime :: Int -> Random -> Natural


module Arithmetic.ContinuedFraction
newtype ContinuedFraction
ContinuedFraction :: (Natural, [Natural]) -> ContinuedFraction
[unContinuedFraction] :: ContinuedFraction -> (Natural, [Natural])
fromNatural :: Natural -> ContinuedFraction
goldenRatio :: ContinuedFraction
naturalLogarithmBase :: ContinuedFraction
convergentsFn :: (Natural -> a) -> (a -> a -> a) -> (a -> a -> a) -> [Natural] -> a -> a -> [a]
numerators :: (Natural -> a) -> (a -> a -> a) -> (a -> a -> a) -> ContinuedFraction -> [a]
denominators :: (Natural -> a) -> (a -> a -> a) -> (a -> a -> a) -> ContinuedFraction -> [a]
convergents :: (Natural -> a) -> (a -> a -> a) -> (a -> a -> a) -> (a -> a -> a) -> ContinuedFraction -> [a]
unstableConvergents :: Eq a => [a] -> [a]
fractionalConvergents :: Fractional a => ContinuedFraction -> [a]
rationalConvergents :: ContinuedFraction -> [Rational]
toDouble :: ContinuedFraction -> Double
fromRealFrac :: RealFrac a => a -> ContinuedFraction
invert :: ContinuedFraction -> Maybe ContinuedFraction
instance GHC.Classes.Eq Arithmetic.ContinuedFraction.ContinuedFraction
instance GHC.Show.Show Arithmetic.ContinuedFraction.ContinuedFraction


module Arithmetic.SquareRoot
floor :: Natural -> Natural
ceiling :: Natural -> Natural
continuedFraction :: Natural -> ContinuedFraction
continuedFractionPeriodic :: Natural -> [Natural]
continuedFractionPeriodicTail :: Natural -> Natural -> [Natural]