MIT licensed and maintained by Andrew Lelechenko

Module documentation for

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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)


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, but offers almost twice faster addition and multiplication, and much less allocations.


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 0.4x 1x 4.5x 6.1x 3.3x 5.0x
Product 0.6x 1x 3.6x 5.4x 3.1x 4.5x
Inversion 0.8x 1x N/A 6.1x N/A 4.1x
Power 0.9x 1x 6.0x 1.8x 1.9x 2.1x

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.


  • Add Storable, Prim and Unbox instances.

  • Add Data.Mod.Word.

  • Initial release
Used by 1 package in nightly-2020-09-09(full list with versions):