Math.NumberTheory.Moduli.Class

Safe modular arithmetic with modulo on type level.

Known modulo

data Mod m

Wrapper for residues modulo m.

Mod 3 :: Mod 10 stands for the class of integers, congruent to 3 modulo 10 (…−17, −7, 3, 13, 23…). The modulo is stored on type level, so it is impossible, for example, to add up by mistake residues with different moduli.

> (3 :: Mod 10) + (4 :: Mod 12)
error: Couldn't match type ‘12’ with ‘10’...
> (3 :: Mod 10) + 8
(1 `modulo` 10)

Note that modulo cannot be negative.

Instances

Eq (Mod m)

Methods

(==) :: Mod m -> Mod m -> Bool

(/=) :: Mod m -> Mod m -> Bool

KnownNat m => Fractional (Mod m)

Beware that division by residue, which is not coprime with the modulo, will result in runtime error. Consider using invertMod instead.

Methods

(/) :: Mod m -> Mod m -> Mod m

recip :: Mod m -> Mod m

fromRational :: Rational -> Mod m

KnownNat m => Num (Mod m)

Methods

(+) :: Mod m -> Mod m -> Mod m

(-) :: Mod m -> Mod m -> Mod m

(*) :: Mod m -> Mod m -> Mod m

negate :: Mod m -> Mod m

abs :: Mod m -> Mod m

signum :: Mod m -> Mod m

fromInteger :: Integer -> Mod m

Ord (Mod m)

Methods

compare :: Mod m -> Mod m -> Ordering

(<) :: Mod m -> Mod m -> Bool

(<=) :: Mod m -> Mod m -> Bool

(>) :: Mod m -> Mod m -> Bool

(>=) :: Mod m -> Mod m -> Bool

max :: Mod m -> Mod m -> Mod m

min :: Mod m -> Mod m -> Mod m

KnownNat m => Show (Mod m)

Methods

showsPrec :: Int -> Mod m -> ShowS

show :: Mod m -> String

showList :: [Mod m] -> ShowS

getVal :: KnownNat m => Mod m -> Integer

The canonical representative of the residue class, always between 0 and m-1 inclusively.

getNatVal :: KnownNat m => Mod m -> Natural

The canonical representative of the residue class, always between 0 and m-1 inclusively.

getMod :: KnownNat m => Mod m -> Integer

Linking type and value levels: extract modulo m as a value.

getNatMod :: KnownNat m => Mod m -> Natural

Linking type and value levels: extract modulo m as a value.

invertMod :: KnownNat m => Mod m -> Maybe (Mod m)

Computes the modular inverse, if the residue is coprime with the modulo.

> invertMod (3 :: Mod 10)
Just (7 `modulo` 10) -- because 3 * 7 = 1 :: Mod 10
> invertMod (4 :: Mod 10)
Nothing

powMod :: (KnownNat m, Integral a) => Mod m -> a -> Mod m

Drop-in replacement for ^, with much better performance.

> powMod (3 :: Mod 10) 4
(1 `modulo` 10)

(^%) :: (KnownNat m, Integral a) => Mod m -> a -> Mod m infixr 8

Infix synonym of powMod.

Unknown modulo

data SomeMod where

This type represents residues with unknown modulo and rational numbers. One can freely combine them in arithmetic expressions, but each operation will spend time on modulo's recalculation:

> 2 `modulo` 10 + 4 `modulo` 15
(1 `modulo` 5)
> 2 `modulo` 10 * 4 `modulo` 15
(3 `modulo` 5)
> 2 `modulo` 10 + fromRational (3 % 7)
(1 `modulo` 10)
> 2 `modulo` 10 * fromRational (3 % 7)
(8 `modulo` 10)

If performance is crucial, it is recommended to extract Mod m for further processing by pattern matching. E. g.,

case modulo n m of
  SomeMod k -> process k -- Here k has type Mod m
  InfMod{}  -> error "impossible"

Constructors

SomeMod :: KnownNat m => Mod m -> SomeMod

InfMod :: Rational -> SomeMod

Instances

Eq SomeMod

Methods

(==) :: SomeMod -> SomeMod -> Bool

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

Fractional SomeMod

Beware that division by residue, which is not coprime with the modulo, will result in runtime error. Consider using invertSomeMod instead.

Methods

(/) :: SomeMod -> SomeMod -> SomeMod

recip :: SomeMod -> SomeMod

fromRational :: Rational -> SomeMod

Num SomeMod

Methods

(+) :: SomeMod -> SomeMod -> SomeMod

(-) :: SomeMod -> SomeMod -> SomeMod

(*) :: SomeMod -> SomeMod -> SomeMod

negate :: SomeMod -> SomeMod

abs :: SomeMod -> SomeMod

signum :: SomeMod -> SomeMod

fromInteger :: Integer -> SomeMod

Show SomeMod

Methods

showsPrec :: Int -> SomeMod -> ShowS

show :: SomeMod -> String

showList :: [SomeMod] -> ShowS

modulo :: Integer -> Natural -> SomeMod infixl 7

Create modular value by representative of residue class and modulo. One can use the result either directly (via functions from Num and Fractional), or deconstruct it by pattern matching. Note that modulo never returns InfMod.

invertSomeMod :: SomeMod -> Maybe SomeMod

Computes the inverse value, if it exists.

> invertSomeMod (3 `modulo` 10)
Just (7 `modulo` 10) -- because 3 * 7 = 1 :: Mod 10
> invertMod (4 `modulo` 10)
Nothing
> invertSomeMod (fromRational (2 % 5))
Just 5 % 2

powSomeMod :: Integral a => SomeMod -> a -> SomeMod

Drop-in replacement for ^, with much better performance. When -O is enabled, there is a rewrite rule, which specialises ^ to powSomeMod.

> powSomeMod (3 `modulo` 10) 4
(1 `modulo` 10)

Re-exported from GHC.TypeNats.Compat

class KnownNat n

This class gives the integer associated with a type-level natural. There are instances of the class for every concrete literal: 0, 1, 2, etc.

Since: 4.7.0.0

Minimal complete definition

natSing