mod

Modular arithmetic, promoting moduli to the type level, with an emphasis on performance. Originally a part of arithmoi package.

> :set -XDataKinds
> 4 + 5 :: Mod 7
(2 `modulo` 7)
> 4 - 5 :: Mod 7
(6 `modulo` 7)
> 4 * 5 :: Mod 7
(6 `modulo` 7)
> 4 / 5 :: Mod 7
(5 `modulo` 7)
> 4 ^ 5 :: Mod 7
(2 `modulo` 7)

Competitors

There are other Haskell packages, employing the very same idea of moduli on the type level, namely modular, modular-arithmetic and finite-field. One can also use finite-typelits, which covers some elementary modular arithmetic as well. Unfortunately, all of them fall behind in terms of performance. Here is a brief comparison:

Discipline	mod	modular	modular-arithmetic	finite-typelits	finite-field
Addition	Fast	Slow	Slow	Slow	Slow
Small (*)	Fast	Slow	Slow	Slow	Slow
Inversion	Fast	N/A	Slow	N/A	Slow
Power	Fast	Slow	Slow	Slow	Slow
Overflows	Safe	Safe	Unsafe	Safe	Safe

• Addition. All competing implementations of the modular addition involve divisions, while mod completely avoids this costly operation. It makes difference even for small numbers; e. g., sum [1..10^7] becomes 5x faster. For larger integers the speed up is even more significant, because the computational complexity of division is not linear.

• Small (*). When a modulo fits a machine word (which is quite a common case on 64-bit architectures), mod implements the modular multiplication as a couple of CPU instructions and neither allocates intermediate arbitrary-precision values, nor calls libgmp at all. For computations like product [1..10^7] this gives a 3x boost to performance in comparison to other libraries.

• Inversion. This package relies on libgmp for modular inversions. Even for small arguments it is about 5x faster than the native implementation of modular inversion in modular-arithmetic.

• Power. This package relies on libgmp for modular exponentiation. Even for small arguments it is about 2x faster than competitors.

• Overflows. At first glance modular-arithmetic is more flexible than mod, because it allows to specify the underlying representation of a modular residue, e. g., Mod Integer 100, Mod Int 100, Mod Word8 100. We argue that this is a dangerous freedom, vulnerable to overflows. For instance, 20 ^ 2 :: Mod Word8 100 returns 44 instead of expected 0. Even less expected is that 50 :: Mod Word8 300 appears to be 6 (remember that type-level numbers are always Natural).

What is the difference between mod and finite-typelits?

mod is specifically designed to represent modular residues for mathematical applications (wrapping-around finite numbers) and provides modular inversion and exponentiation.

The main focus of finite-typelits is on non-wrapping-around finite numbers, like indices of arrays in vector-sized. It features a Num instance only for the sake of overloading numeric literals. There is no lawful way to define Num except modular arithmetic, but from finite-typelits viewpoint this is a by-product.

Citius, altius, fortius!

If you are looking for an ultimate performance and your moduli fit into Word, try Data.Mod.Word, which is a drop-in replacement of Data.Mod, offering better performance and much less allocations.

Benchmarks

Here are some relative benchmarks (less is better), which can be reproduced by running cabal bench.

Discipline	Data.Mod.Word	Data.Mod	modular	modular-arithmetic	finite-typelits	finite-field
Sum	0.25x	1x	11.4x	5.7x	8.9x	8.6x
Product	0.95x	1x	9.6x	4.8x	7.0x	7.0x
Inversion	0.95x	1x	N/A	2.6x	N/A	3.0x
Power	0.90x	1x	6.9x	3.8x	5.0x	4.9x

What's next?

This package was cut out of arithmoi to provide a modular arithmetic with a light dependency footprint. This goal certainly limits the scope of API to the bare minimum. If you need more advanced tools (the Chinese remainder theorem, cyclic groups, modular equations, etc.) please refer to Math.NumberTheory.Moduli.