Data.Mod.Word · mod-0.2.0.1 · horizon-platform

newtypenewtype Mod (m :: Nat)

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

Example2 expressions

>>> :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} .

Instances24Bounded, Enum, Eq, Fractional, Num, Ord, …

Vector Vector (Mod m)Defined in mod-0.2.0.1 · Data.Mod.Word
No validation checks are performed; reading untrusted data may corrupt internal invariants.
MVector MVector (Mod m)Defined in mod-0.2.0.1 · Data.Mod.Word
No validation checks are performed; reading untrusted data may corrupt internal invariants.
KnownNat m => Bounded (Mod m)Defined in mod-0.2.0.1 · Data.Mod.Word
KnownNat m => Enum (Mod m)Defined in mod-0.2.0.1 · Data.Mod.Word
Eq (Mod m)Defined in mod-0.2.0.1 · Data.Mod.Word
KnownNat m => Fractional (Mod m)Defined in mod-0.2.0.1 · Data.Mod.Word
Division by a residue, which is not coprime with the modulus, throws DivideByZero. Consider using invertMod for non-prime moduli.
KnownNat m => Num (Mod m)Defined in mod-0.2.0.1 · Data.Mod.Word
Ord (Mod m)Defined in mod-0.2.0.1 · Data.Mod.Word
KnownNat m => Read (Mod m)Defined in mod-0.2.0.1 · Data.Mod.Word
Wrapping behaviour, similar to the existing instance Read Int.
KnownNat m => Real (Mod m)Defined in mod-0.2.0.1 · Data.Mod.Word
Show (Mod m)Defined in mod-0.2.0.1 · Data.Mod.Word
Generic (Mod m)Defined in mod-0.2.0.1 · Data.Mod.Word
Storable (Mod m)Defined in mod-0.2.0.1 · Data.Mod.Word
No validation checks are performed; reading untrusted data may corrupt internal invariants.
NFData (Mod m)Defined in mod-0.2.0.1 · Data.Mod.Word
Prim (Mod m)Defined in mod-0.2.0.1 · Data.Mod.Word
No validation checks are performed; reading untrusted data may corrupt internal invariants.
Unbox (Mod m)Defined in mod-0.2.0.1 · Data.Mod.Word
No validation checks are performed; reading untrusted data may corrupt internal invariants.
KnownNat m => Euclidean (Mod m)Defined in mod-0.2.0.1 · Data.Mod.Word
Mod m is not even an integral domain for composite m, much less a Euclidean domain.
The instance is lawful only for prime m, otherwise we try to do our best: quot x y returns any z such that x == y * z, rem is not always 0, and both can throw DivideByZero.
KnownNat m => Field (Mod m)Defined in mod-0.2.0.1 · Data.Mod.Word
Mod m is not even an integral domain for composite m, much less a field.
The instance is lawful only for prime m, otherwise division by a residue, which is not coprime with the modulus, throws DivideByZero. Consider using invertMod for non-prime moduli.
KnownNat m => GcdDomain (Mod m)Defined in mod-0.2.0.1 · Data.Mod.Word
Mod m is not even an integral domain for composite m, much less a GCD domain. However, gcd and lcm are still meaningful even for composite m, corresponding to a sum and an intersection of ideals.
The instance is lawful only for prime m, otherwise divide x y tries to return any Just z such that x == y * z.
KnownNat m => Semiring (Mod m)Defined in mod-0.2.0.1 · Data.Mod.Word
KnownNat m => Ring (Mod m)Defined in mod-0.2.0.1 · Data.Mod.Word
type Rep (Mod m) = D1 ('MetaData "Mod" "Data.Mod.Word" "mod-0.2.0.1-9dF6mWbkkfOFPRbP3H3cXx" 'True) (C1 ('MetaCons "Mod" 'PrefixI 'True) (S1 ('MetaSel ('Just "unMod") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 Word)))Defined in mod-0.2.0.1 · Data.Mod.Word
data MVector s (Mod m)
- MV_Mod (MVector s Word)
Defined in mod-0.2.0.1 · Data.Mod.Word
data Vector (Mod m)
- V_Mod (Vector Word)
Defined in mod-0.2.0.1 · Data.Mod.Word

valueunMod :: Mod m -> Word

#

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

Example2 expressions

>>> :set -XDataKinds>>> -1 :: Mod 109

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

#

If an argument is coprime with the modulus, return its modular inverse. Otherwise return Nothing.

Example3 expressions

>>> :set -XDataKinds>>> invertMod 3 :: Mod 10 -- 3 * 7 = 21 ≡ 1 (mod 10)Just 7>>> invertMod 4 :: Mod 10 -- 4 and 10 are not coprimeNothing

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

#

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.

Example4 expressions

>>> :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