integer-gmp-1.0.2.0: Integer library based on GMP

Copyright(c) Herbert Valerio Riedel 2014
LicenseBSD3
Maintainerghc-devs@haskell.org
Stabilityprovisional
Portabilitynon-portable (GHC Extensions)
Safe HaskellNone
LanguageHaskell2010

GHC.Integer.GMP.Internals

Contents

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

The Integer type

data Integer Source #

Invariant: Jn# and Jp# are used iff value doesn't fit in S#

Useful properties resulting from the invariants:

Constructors

S# !Int#

iff value in [minBound::Int, maxBound::Int] range

Jp# !BigNat

iff value in ]maxBound::Int, +inf[ range

Jn# !BigNat

iff value in ]-inf, minBound::Int[ range

Instances
Eq Integer Source # 
Instance details

Defined in GHC.Integer.Type

Methods

(==) :: Integer -> Integer -> Bool

(/=) :: Integer -> Integer -> Bool

Ord Integer Source # 
Instance details

Defined in GHC.Integer.Type

isValidInteger# :: Integer -> Int# Source #

Test whether all internal invariants are satisfied by Integer value

Returns 1# if valid, 0# otherwise.

This operation is mostly useful for test-suites and/or code which constructs Integer values directly.

Basic Integer operations

Additional Integer operations

bitInteger :: Int# -> Integer Source #

Integer for which only n-th bit is set. Undefined behaviour for negative n values.

popCountInteger :: Integer -> Int# Source #

Count number of set bits. For negative arguments returns negative population count of negated argument.

gcdInteger :: Integer -> Integer -> Integer Source #

Compute greatest common divisor.

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

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 Source #

Compute least common multiple.

powModInteger :: Integer -> Integer -> Integer -> Integer Source #

"powModInteger b e m" computes base b raised to exponent e modulo abs(m).

Negative exponents are supported if an inverse modulo m exists.

Warning: 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.

Future versions of integer_gmp may not support negative e values anymore.

Since: 0.5.1.0

powModSecInteger :: Integer -> Integer -> Integer -> Integer Source #

"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: 1.0.2.0

recipModInteger :: Integer -> Integer -> Integer Source #

"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.

Since: 0.5.1.0

Additional conversion operations to Integer

The BigNat type

data BigNat Source #

Type representing raw arbitrary-precision Naturals

This is common type used by Natural and Integer. As this type consists of a single constructor wrapping a ByteArray# it can be unpacked.

Essential invariants:

  • ByteArray# size is an exact multiple of Word# size
  • limbs are stored in least-significant-limb-first order,
  • the most-significant limb must be non-zero, except for
  • 0 which is represented as a 1-limb.

Constructors

BN# ByteArray# 
Instances
Eq BigNat Source # 
Instance details

Defined in GHC.Integer.Type

Methods

(==) :: BigNat -> BigNat -> Bool

(/=) :: BigNat -> BigNat -> Bool

Ord BigNat Source # 
Instance details

Defined in GHC.Integer.Type

type GmpLimb = Word Source #

Type representing a GMP Limb

type GmpSize = Int Source #

Count of GmpLimbs, must be positive (unless specified otherwise).

isValidBigNat# :: BigNat -> Int# Source #

Test whether all internal invariants are satisfied by BigNat value

Returns 1# if valid, 0# otherwise.

This operation is mostly useful for test-suites and/or code which constructs Integer values directly.

sizeofBigNat# :: BigNat -> GmpSize# Source #

Return number of limbs contained in BigNat.

zeroBigNat :: BigNat Source #

CAF representing the value 0 :: BigNat

oneBigNat :: BigNat Source #

CAF representing the value 1 :: BigNat

nullBigNat :: BigNat Source #

Special 0-sized bigNat returned in case of arithmetic underflow

This is currently only returned by the following operations:

Other operations such as quotBigNat may return nullBigNat as well as a dummy/place-holder value instead of undefined since we can't throw exceptions. But that behaviour should not be relied upon.

NB: isValidBigNat# nullBigNat is false

Conversions to/from BigNat

byteArrayToBigNat# :: ByteArray# -> GmpSize# -> BigNat Source #

Construct BigNat from existing ByteArray# containing n GmpLimbs in least-significant-first order.

If possible ByteArray#, will be used directly (i.e. shared without cloning the ByteArray# into a newly allocated one)

Note: size parameter (times sizeof(GmpLimb)) must be less or equal to its sizeofByteArray#.

wordToBigNat :: Word# -> BigNat Source #

Construct 1-limb BigNat from Word#

wordToBigNat2 :: Word# -> Word# -> BigNat Source #

Construct BigNat from 2 limbs. The first argument is the most-significant limb.

indexBigNat# :: BigNat -> GmpSize# -> GmpLimb# Source #

Extract n-th (0-based) limb in BigNat. n must be less than size as reported by sizeofBigNat#.

BigNat arithmetic operations

minusBigNat :: BigNat -> BigNat -> BigNat Source #

Returns nullBigNat (see isNullBigNat#) in case of underflow

minusBigNatWord :: BigNat -> GmpLimb# -> BigNat Source #

Returns nullBigNat (see isNullBigNat#) in case of underflow

quotRemBigNat :: BigNat -> BigNat -> (#BigNat, BigNat#) Source #

If divisor is zero, (# nullBigNat, nullBigNat #) is returned

quotRemBigNatWord :: BigNat -> GmpLimb# -> (#BigNat, GmpLimb##) Source #

Note: Result of div/0 undefined

remBigNatWord :: BigNat -> GmpLimb# -> Word# Source #

div/0 not checked

powModBigNat :: BigNat -> BigNat -> BigNat -> BigNat Source #

Version of powModInteger operating on BigNats

Since: 1.0.0.0

powModBigNatWord :: BigNat -> BigNat -> GmpLimb# -> GmpLimb# Source #

Version of powModInteger for Word#-sized moduli

Since: 1.0.0.0

recipModBigNat :: BigNat -> BigNat -> BigNat Source #

Version of recipModInteger operating on BigNats

Since: 1.0.0.0

BigNat logic operations

bitBigNat :: Int# -> BigNat Source #

Specialised version of

bitBigNat = shiftLBigNat (wordToBigNat 1##)

avoiding a few redundant allocations

BigNat comparison predicates

isZeroBigNat :: BigNat -> Bool Source #

Test if BigNat value is equal to zero.

isNullBigNat# :: BigNat -> Int# Source #

Test for special 0-sized BigNat representing underflows.

Miscellaneous GMP-provided operations

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

Compute greatest common divisor.

Warning: result may become negative if (at least) one argument is minBound

gcdWord :: Word# -> Word# -> Word# Source #

Compute greatest common divisor.

Since: 1.0.0.0

powModWord :: GmpLimb# -> GmpLimb# -> GmpLimb# -> GmpLimb# Source #

Version of powModInteger operating on Word#s

Since: 1.0.0.0

recipModWord :: GmpLimb# -> GmpLimb# -> GmpLimb# Source #

Version of recipModInteger operating on Word#s

Since: 1.0.0.0

Primality tests

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

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

testPrimeBigNat :: BigNat -> Int# -> Int# Source #

Version of testPrimeInteger operating on BigNats

Since: 1.0.0.0

testPrimeWord# :: GmpLimb# -> Int# -> Int# Source #

Version of testPrimeInteger operating on Word#s

Since: 1.0.0.0

nextPrimeInteger :: Integer -> Integer Source #

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

nextPrimeBigNat :: BigNat -> BigNat Source #

Version of nextPrimeInteger operating on BigNats

Since: 1.0.0.0

nextPrimeWord# :: GmpLimb# -> GmpLimb# Source #

Version of nextPrimeInteger operating on Word#s

Since: 1.0.0.0

Import/export functions

Compute size of serialisation

sizeInBaseBigNat :: BigNat -> Int# -> Word# Source #

Version of sizeInBaseInteger operating on BigNat

Since: 1.0.0.0

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

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

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

sizeInBaseWord# :: Word# -> Int# -> Word# Source #

Version of sizeInBaseInteger operating on Word#

Since: 1.0.0.0

Export

exportIntegerToAddr :: Integer -> Addr# -> Int# -> IO Word Source #

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

exportIntegerToAddr i addr e

See description of exportIntegerToMutableByteArray for more details.

Since: 1.0.0.0

exportWordToAddr :: Word -> Addr# -> Int# -> IO Word Source #

Version of exportIntegerToAddr operating on Words.

exportIntegerToMutableByteArray :: Integer -> MutableByteArray# RealWorld -> Word# -> Int# -> IO Word Source #

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

The call

exportIntegerToMutableByteArray i mba offset msbf

writes

  • the Integer i
  • into the MutableByteArray# mba starting at offset
  • with most significant byte first if msbf is 1# or least significant byte first if msbf is 0#, 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 msbf to call mpz_export().

Since: 1.0.0.0

Import

importIntegerFromAddr :: Addr# -> Word# -> Int# -> IO Integer Source #

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

importIntegerFromAddr addr size msbf

See description of importIntegerFromByteArray for more details.

Since: 1.0.0.0

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

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

The call

importIntegerFromByteArray ba offset size msbf

reads

  • size bytes from the ByteArray# ba starting at offset
  • with most significant byte first if msbf is 1# or least significant byte first if msbf is 0#, and
  • returns a new Integer

Since: 1.0.0.0