integer-gmp-0.5.1.0: Integer library based on GMP

Safe Haskell | None
Language | Haskell2010

GHC.Integer.GMP.Internals

Contents

- The `Integer` type
- Number theoretic functions
- Exponentiation functions
- Import/export functions

Description

This modules provides access to the `Integer` constructors and exposes some highly optimized GMP-operations.

Note that since `integer-gmp` does not depend on `base`, error reporting via exceptions, `error`, or `undefined` is not available. Instead, the low-level functions will crash the runtime if called with invalid arguments.

See also GHC Commentary: Libraries/Integer.

Synopsis

- data Integer
  - = S# Int#
  - | J# Int# ByteArray#
- gcdInt :: Int# -> Int# -> Int#
- gcdInteger :: Integer -> Integer -> Integer
- gcdExtInteger :: Integer -> Integer -> (#Integer, Integer#)
- lcmInteger :: Integer -> Integer -> Integer
- nextPrimeInteger :: Integer -> Integer
- testPrimeInteger :: Integer -> Int# -> Int#
- powInteger :: Integer -> Word# -> Integer
- powModInteger :: Integer -> Integer -> Integer -> Integer
- powModSecInteger :: Integer -> Integer -> Integer -> Integer
- recipModInteger :: Integer -> Integer -> Integer
- sizeInBaseInteger :: Integer -> Int# -> Word#
- importIntegerFromByteArray :: ByteArray# -> Word# -> Word# -> Int# -> Integer
- importIntegerFromAddr :: Addr# -> Word# -> Int# -> State# s -> (#State# s, Integer#)
- exportIntegerToMutableByteArray :: Integer -> MutableByteArray# s -> Word# -> Int# -> State# s -> (#State# s, Word##)
- exportIntegerToAddr :: Integer -> Addr# -> Int# -> State# s -> (#State# s, Word##)

# The `Integer` type

data Integer

Arbitrary-precision integers.

Constructors

S# Int# | "small" integers fitting into an `Int#`
J# Int# ByteArray# | "big" integers represented as GMP's `mpz_t` structure. The `Int#` field corresponds to `mpz_t`'s `_mp_size` field, which encodes the sign and the number of limbs stored in the `ByteArray#` field (i.e. `mpz_t`'s `_mp_d` field). Note: The `ByteArray#` may have been over-allocated, and thus larger than the size denoted by the `Int#` field. This representation tries to avoid using the GMP number representation for small integers that fit into a native `Int#`. This allows to reduce (or at least defer) calling into GMP for operations whose results remain in the `Int#`-domain. Note: It does not constitute a violation of invariants to represent an integer which would fit into an `Int#` with the `J#`-constructor. For instance, the value `0` has (only) two valid representations, either ``S#` 0#` or ``J#` 0 _`.

Instances

Eq Integer |
Ord Integer |

# Number theoretic functions

gcdInt :: Int# -> Int# -> Int#

Compute greatest common divisor.

gcdInteger :: Integer -> Integer -> Integer

Compute greatest common divisor.

gcdExtInteger :: Integer -> Integer -> (#Integer, Integer#)

Extended euclidean algorithm.

For `a` and `b`, compute their greatest common divisor `g` and the coefficient `s` satisfying `as + bt = g`.

Since: 0.5.1.0

lcmInteger :: Integer -> Integer -> Integer

Compute least common multiple.

nextPrimeInteger :: Integer -> Integer

Compute next prime greater than `n` probalistically.

According to the GMP documentation, the underlying function `mpz_nextprime()` "uses a probabilistic algorithm to identify primes. For practical purposes it's adequate, the chance of a composite passing will be extremely small."

Since: 0.5.1.0

testPrimeInteger :: Integer -> Int# -> Int#

Probalistic Miller-Rabin primality test.

"``testPrimeInteger` n k`" determines whether `n` is prime and returns one of the following results:

- `2#` is returned if `n` is definitely prime,
- `1#` if `n` is a probable prime, or
- `0#` if `n` is definitely not a prime.

The `k` argument controls how many test rounds are performed for determining a probable prime. For more details, see GMP documentation for `mpz_probab_prime_p()`.

Since: 0.5.1.0

# Exponentiation functions

powInteger :: Integer -> Word# -> Integer

"``powInteger` b e`" computes base `b` raised to exponent `e`.

Since: 0.5.1.0

powModInteger :: Integer -> Integer -> Integer -> Integer

"``powModInteger` b e m`" computes base `b` raised to exponent `e` modulo `m`.

Negative exponents are supported if an inverse modulo `m` exists. It's advised to avoid calling this primitive with negative exponents unless it is guaranteed the inverse exists, as failure to do so will likely cause program abortion due to a divide-by-zero fault. See also `recipModInteger`.

Since: 0.5.1.0

powModSecInteger :: Integer -> Integer -> Integer -> Integer

"``powModSecInteger` b e m`" computes base `b` raised to exponent `e` modulo `m`. It is required that `e > 0` and `m` is odd.

This is a "secure" variant of `powModInteger` using the `mpz_powm_sec()` function which is designed to be resilient to side channel attacks and is therefore intended for cryptographic applications.

This primitive is only available when the underlying GMP library supports it (GMP >= 5). Otherwise, it internally falls back to ``powModInteger``, and a warning will be emitted when used.

Since: 0.5.1.0

recipModInteger :: Integer -> Integer -> Integer

"``recipModInteger` x m`" computes the inverse of `x` modulo `m`. If the inverse exists, the return value `y` will satisfy `0 < y < abs(m)`, otherwise the result is `0`.

Note: The implementation exploits the undocumented property of `mpz_invert()` to not mangle the result operand (which is initialized to 0) in case of non-existence of the inverse.

Since: 0.5.1.0

# Import/export functions

sizeInBaseInteger :: Integer -> Int# -> Word#

Compute number of digits (without sign) in given `base`.

It's recommended to avoid calling `sizeInBaseInteger` for small integers as this function would currently convert those to big integers in order to call `mpz_sizeinbase()`.

This function wraps `mpz_sizeinbase()` which has some implementation pecularities to take into account:

- "``sizeInBaseInteger` 0 base = 1`" (see also comment in `exportIntegerToMutableByteArray`).
- This function is only defined if `base >= 2#` and `base <= 256#` (Note: the documentation claims that only `base <= 62#` is supported, however the actual implementation supports up to base 256).
- If `base` is a power of 2, the result will be exact. In other cases (e.g. for `base = 10#`), the result may be 1 digit too large sometimes.
- "``sizeInBaseInteger` i 2#`" can be used to determine the most significant bit of `i`.

Since: 0.5.1.0

importIntegerFromByteArray :: ByteArray# -> Word# -> Word# -> Int# -> Integer

Read `Integer` (without sign) from byte-array in base-256 representation.

The call

    importIntegerFromByteArray ba offset size order

reads

- `size` bytes from the `ByteArray#` `ba` starting at `offset`
- with most significant byte first if `order` is `1#` or least significant byte first if `order` is `-1#`, and
- returns a new `Integer`

Since: 0.5.1.0

importIntegerFromAddr :: Addr# -> Word# -> Int# -> State# s -> (#State# s, Integer#)

Read `Integer` (without sign) from memory location at `addr` in base-256 representation.

    importIntegerFromAddr addr size order

See description of `importIntegerFromByteArray` for more details.

Since: 0.5.1.0

exportIntegerToMutableByteArray :: Integer -> MutableByteArray# s -> Word# -> Int# -> State# s -> (#State# s, Word##)

Dump `Integer` (without sign) to mutable byte-array in base-256 representation.

The call

    exportIntegerToMutableByteArray i mba offset order

writes

- the `Integer` `i`
- into the `MutableByteArray#` `mba` starting at `offset`
- with most significant byte first if `order` is `1#` or least significant byte first if `order` is `-1#`, and
- returns number of bytes written.

Use "``sizeInBaseInteger` i 256#`" to compute the exact number of bytes written in advance for `i /= 0`. In case of `i == 0`, `exportIntegerToMutableByteArray` will write and report zero bytes written, whereas `sizeInBaseInteger` report one byte.

It's recommended to avoid calling `exportIntegerToMutableByteArray` for small integers as this function would currently convert those to big integers in order to call `mpz_export()`.

Since: 0.5.1.0

exportIntegerToAddr :: Integer -> Addr# -> Int# -> State# s -> (#State# s, Word##)

Dump `Integer` (without sign) to `addr` in base-256 representation.

    exportIntegerToAddr addr o e

See description of `exportIntegerToMutableByteArray` for more details.

Since: 0.5.1.0