| Copyright | (c) 2017-2022 Andrew Lelechenko |
|---|---|
| License | MIT |
| Maintainer | Andrew Lelechenko <andrew.lelechenko@gmail.com> |
| Safe Haskell | Safe-Inferred |
| Language | Haskell2010 |
Data.Mod.Word
Description
Modular arithmetic, promoting moduli to the type level, with an emphasis on performance. Originally part of the arithmoi package.
This module supports only moduli, which fit into Word.
Use the (slower) Data.Mod module for handling arbitrary-sized moduli.
Documentation
This data type represents integers modulo m, equipped with useful instances.
For example, 3 :: Mod 10 stands for the class of integers
congruent to \( 3 \bmod 10 \colon \ldots {−17}, −7, 3, 13, 23 \ldots \)
>>>:set -XDataKinds>>>3 + 8 :: Mod 10 -- 3 + 8 = 11 ≡ 1 (mod 10)1
Note: Mod 0 has no inhabitants, eventhough \( \mathbb{Z}/0\mathbb{Z} \) is technically isomorphic to \( \mathbb{Z} \).
Instances
| Vector Vector (Mod m) # | No validation checks are performed; reading untrusted data may corrupt internal invariants. |
Defined in Data.Mod.Word Methods basicUnsafeFreeze :: Mutable Vector s (Mod m) -> ST s (Vector (Mod m)) # basicUnsafeThaw :: Vector (Mod m) -> ST s (Mutable Vector s (Mod m)) # basicLength :: Vector (Mod m) -> Int # basicUnsafeSlice :: Int -> Int -> Vector (Mod m) -> Vector (Mod m) # basicUnsafeIndexM :: Vector (Mod m) -> Int -> Box (Mod m) # basicUnsafeCopy :: Mutable Vector s (Mod m) -> Vector (Mod m) -> ST s () # | |
| MVector MVector (Mod m) # | No validation checks are performed; reading untrusted data may corrupt internal invariants. |
Defined in Data.Mod.Word Methods basicLength :: MVector s (Mod m) -> Int # basicUnsafeSlice :: Int -> Int -> MVector s (Mod m) -> MVector s (Mod m) # basicOverlaps :: MVector s (Mod m) -> MVector s (Mod m) -> Bool # basicUnsafeNew :: Int -> ST s (MVector s (Mod m)) # basicInitialize :: MVector s (Mod m) -> ST s () # basicUnsafeReplicate :: Int -> Mod m -> ST s (MVector s (Mod m)) # basicUnsafeRead :: MVector s (Mod m) -> Int -> ST s (Mod m) # basicUnsafeWrite :: MVector s (Mod m) -> Int -> Mod m -> ST s () # basicClear :: MVector s (Mod m) -> ST s () # basicSet :: MVector s (Mod m) -> Mod m -> ST s () # basicUnsafeCopy :: MVector s (Mod m) -> MVector s (Mod m) -> ST s () # basicUnsafeMove :: MVector s (Mod m) -> MVector s (Mod m) -> ST s () # basicUnsafeGrow :: MVector s (Mod m) -> Int -> ST s (MVector s (Mod m)) # | |
| Storable (Mod m) # | No validation checks are performed; reading untrusted data may corrupt internal invariants. |
Defined in Data.Mod.Word Methods peekElemOff :: Ptr (Mod m) -> Int -> IO (Mod m) pokeElemOff :: Ptr (Mod m) -> Int -> Mod m -> IO () peekByteOff :: Ptr b -> Int -> IO (Mod m) pokeByteOff :: Ptr b -> Int -> Mod m -> IO () | |
| KnownNat m => Bounded (Mod m) # | |
Defined in Data.Mod.Word | |
| KnownNat m => Enum (Mod m) # | |
| Generic (Mod m) # | |
| KnownNat m => Num (Mod m) # | |
| KnownNat m => Read (Mod m) # | Wrapping behaviour, similar to
the existing |
| KnownNat m => Fractional (Mod m) # | Division by a residue, which is not
coprime
with the modulus, throws |
Defined in Data.Mod.Word | |
| KnownNat m => Real (Mod m) # | |
Defined in Data.Mod.Word Methods toRational :: Mod m -> Rational | |
| Show (Mod m) # | |
| NFData (Mod m) # | |
Defined in Data.Mod.Word | |
| Eq (Mod m) # | |
| Ord (Mod m) # | |
| Prim (Mod m) # | No validation checks are performed; reading untrusted data may corrupt internal invariants. |
Defined in Data.Mod.Word Methods sizeOfType# :: Proxy (Mod m) -> Int# # alignmentOfType# :: Proxy (Mod m) -> Int# # alignment# :: Mod m -> Int# # indexByteArray# :: ByteArray# -> Int# -> Mod m # readByteArray# :: MutableByteArray# s -> Int# -> State# s -> (# State# s, Mod m #) # writeByteArray# :: MutableByteArray# s -> Int# -> Mod m -> State# s -> State# s # setByteArray# :: MutableByteArray# s -> Int# -> Int# -> Mod m -> State# s -> State# s # indexOffAddr# :: Addr# -> Int# -> Mod m # readOffAddr# :: Addr# -> Int# -> State# s -> (# State# s, Mod m #) # writeOffAddr# :: Addr# -> Int# -> Mod m -> State# s -> State# s # setOffAddr# :: Addr# -> Int# -> Int# -> Mod m -> State# s -> State# s # | |
| KnownNat m => Euclidean (Mod m) # |
The instance is lawful only for
prime |
| KnownNat m => Field (Mod m) # |
The instance is lawful only for
prime |
Defined in Data.Mod.Word | |
| KnownNat m => GcdDomain (Mod m) # |
The instance is lawful only for
prime |
| KnownNat m => Ring (Mod m) # | |
Defined in Data.Mod.Word | |
| KnownNat m => Semiring (Mod m) # | |
| Unbox (Mod m) # | No validation checks are performed; reading untrusted data may corrupt internal invariants. |
Defined in Data.Mod.Word | |
| newtype MVector s (Mod m) # | |
Defined in Data.Mod.Word | |
| type Rep (Mod m) # | |
Defined in Data.Mod.Word type Rep (Mod m) = D1 ('MetaData "Mod" "Data.Mod.Word" "mod-0.2.0.1-AFA22eBpRMGzYLceit4tM" 'True) (C1 ('MetaCons "Mod" 'PrefixI 'True) (S1 ('MetaSel ('Just "unMod") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 Word))) | |
| newtype Vector (Mod m) # | |
Defined in Data.Mod.Word | |
The canonical representative of the residue class, always between 0 and \( m - 1 \) (inclusively).
>>>:set -XDataKinds>>>-1 :: Mod 109
invertMod :: KnownNat m => Mod m -> Maybe (Mod m) #
If an argument is
coprime
with the modulus, return its modular inverse.
Otherwise return Nothing.
>>>:set -XDataKinds>>>invertMod 3 :: Mod 10 -- 3 * 7 = 21 ≡ 1 (mod 10)Just 7>>>invertMod 4 :: Mod 10 -- 4 and 10 are not coprimeNothing
(^%) :: (KnownNat m, Integral a) => Mod m -> a -> Mod m infixr 8 #
Drop-in replacement for ^ with a bit better performance.
Negative powers are allowed, but may throw DivideByZero, if an argument
is not coprime with the modulus.
>>>:set -XDataKinds>>>3 ^% 4 :: Mod 10 -- 3 ^ 4 = 81 ≡ 1 (mod 10)1>>>3 ^% (-1) :: Mod 10 -- 3 * 7 = 21 ≡ 1 (mod 10)7>>>4 ^% (-1) :: Mod 10 -- 4 and 10 are not coprime(*** Exception: divide by zero