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

Useful properties resulting from the invariants:

• abs (S# _) <= abs (Jp# _)

• abs (S# _) <  abs (Jn# _)

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.

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

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

Compute greatest common divisor.

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

Compute least common multiple.

Square Integer

"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 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 TODO

"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

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.

Type representing a GMP Limb

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

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.

Return number of limbs contained in BigNat.

CAF representing the value 0 :: BigNat

CAF representing the value 1 :: BigNat

Special 0-sized bigNat returned in case of arithmetic underflow

This is currently only returned by the following operations:

• minusBigNat
• minusBigNatWord

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

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#.

Construct 1-limb BigNat from Word#

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

Equivalent to word2Int# . bigNatToWord

Same as indexBigNat# bn 0#

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

Returns nullBigNat (see isNullBigNat#) in case of underflow

Returns nullBigNat (see isNullBigNat#) in case of underflow

Square BigNat

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

Note: Result of div/0 undefined

div/0 not checked

Version of powModInteger operating on BigNats

Since: 1.0.0.0

Version of powModInteger for Word#-sized moduli

Since: 1.0.0.0

Version of recipModInteger operating on BigNats

Since: 1.0.0.0

Specialised version of

bitBigNat = shiftLBigNat (wordToBigNat 1##)

avoiding a few redundant allocations

Test if BigNat value is equal to zero.

Test for special 0-sized BigNat representing underflows.

Compute greatest common divisor.

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

Compute greatest common divisor.

Since: 1.0.0.0

Version of powModInteger operating on Word#s

Since: 1.0.0.0

Version of recipModInteger operating on Word#s

Since: 1.0.0.0

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

Version of testPrimeInteger operating on BigNats

Since: 1.0.0.0

Version of testPrimeInteger operating on Word#s

Since: 1.0.0.0

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

Version of nextPrimeInteger operating on BigNats

Since: 1.0.0.0

Version of nextPrimeInteger operating on Word#s

Since: 1.0.0.0

Version of sizeInBaseInteger operating on BigNat

Since: 1.0.0.0

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

Version of sizeInBaseInteger operating on Word#

Since: 1.0.0.0

Version of exportIntegerToAddr operating on BigNats.

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

Version of exportIntegerToAddr operating on Words.

Version of exportIntegerToMutableByteArray operating on BigNats.

Since: 1.0.0.0

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

Version of exportIntegerToMutableByteArray operating on Words.

Since: 1.0.0.0

Version of importIntegerFromAddr constructing a BigNat

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

Version of importIntegerFromByteArray constructing a BigNat

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