Copyright | (c) Herbert Valerio Riedel 2014 |
---|---|
License | BSD3 |
Maintainer | [email protected] |
Stability | provisional |
Portability | non-portable (GHC Extensions) |
Safe Haskell | None |
Language | Haskell2010 |
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.
Invariant: Jn#
and Jp#
are used iff value doesn't fit in S#
Useful properties resulting from the invariants:
S# !Int# | |
Jp# !BigNat | iff value in |
Jn# !BigNat | iff value in |
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.
Integer
operationsmodule GHC.Integer
Integer
operationsbitInteger :: 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: integer-gmp-0.5.1.0
lcmInteger :: Integer -> Integer -> Integer Source
Compute least common multiple.
sqrInteger :: Integer -> Integer Source
Square Integer
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: integer-gmp-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: integer-gmp-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: integer-gmp-0.5.1.0
Integer
wordToNegInteger :: Word# -> Integer Source
bigNatToInteger :: BigNat -> Integer Source
bigNatToNegInteger :: BigNat -> Integer 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#
size0
which is represented as a 1-limb.BN# ByteArray# |
Type representing a GMP Limb
Count of GmpLimb
s, 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
.
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:
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
BigNat
byteArrayToBigNat# :: ByteArray# -> GmpSize# -> BigNat Source
Construct BigNat
from existing ByteArray#
containing n GmpLimb
s 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.
bigNatToInt :: BigNat -> Int# Source
Equivalent to word2Int# . bigNatToWord
bigNatToWord :: BigNat -> Word# Source
Same as indexBigNat# bn 0#
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 operationsplusBigNat :: BigNat -> BigNat -> BigNat Source
plusBigNatWord :: BigNat -> GmpLimb# -> BigNat Source
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
timesBigNat :: BigNat -> BigNat -> BigNat Source
timesBigNatWord :: BigNat -> GmpLimb# -> BigNat Source
sqrBigNat :: BigNat -> BigNat Source
Square BigNat
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
quotBigNatWord :: BigNat -> GmpLimb# -> BigNat Source
quotBigNat :: BigNat -> BigNat -> BigNat Source
remBigNat :: BigNat -> BigNat -> BigNat Source
remBigNatWord :: BigNat -> GmpLimb# -> Word# Source
div/0 not checked
gcdBigNat :: BigNat -> BigNat -> BigNat Source
gcdBigNatWord :: BigNat -> Word# -> Word# Source
powModBigNat :: BigNat -> BigNat -> BigNat -> BigNat Source
Version of powModInteger
operating on BigNat
s
Since: integer-gmp-1.0.0.0
powModBigNatWord :: BigNat -> BigNat -> GmpLimb# -> GmpLimb# Source
Version of powModInteger
for Word#
-sized moduli
Since: integer-gmp-1.0.0.0
recipModBigNat :: BigNat -> BigNat -> BigNat Source
Version of recipModInteger
operating on BigNat
s
Since: integer-gmp-1.0.0.0
BigNat
logic operationsshiftRBigNat :: BigNat -> Int# -> BigNat Source
shiftLBigNat :: BigNat -> Int# -> BigNat Source
testBitBigNat :: BigNat -> Int# -> Bool Source
clearBitBigNat :: BigNat -> Int# -> BigNat Source
complementBitBigNat :: BigNat -> Int# -> BigNat Source
setBitBigNat :: BigNat -> Int# -> BigNat Source
andBigNat :: BigNat -> BigNat -> BigNat Source
xorBigNat :: BigNat -> BigNat -> BigNat Source
popCountBigNat :: BigNat -> Int# Source
orBigNat :: BigNat -> BigNat -> BigNat Source
bitBigNat :: Int# -> BigNat Source
Specialised version of
bitBigNat = shiftLBigNat (wordToBigNat 1##)
avoiding a few redundant allocations
BigNat
comparison predicatesisZeroBigNat :: BigNat -> Bool Source
Test if BigNat
value is equal to zero.
isNullBigNat# :: BigNat -> Int# Source
Test for special 0-sized BigNat
representing underflows.
compareBigNatWord :: BigNat -> GmpLimb# -> Ordering Source
compareBigNat :: BigNat -> BigNat -> Ordering Source
eqBigNatWord :: BigNat -> GmpLimb# -> Bool Source
eqBigNatWord# :: BigNat -> GmpLimb# -> Int# Source
eqBigNat :: BigNat -> BigNat -> Bool Source
eqBigNat# :: BigNat -> BigNat -> Int# Source
gtBigNatWord# :: BigNat -> GmpLimb# -> Int# Source
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: integer-gmp-1.0.0.0
powModWord :: GmpLimb# -> GmpLimb# -> GmpLimb# -> GmpLimb# Source
Version of powModInteger
operating on Word#
s
Since: integer-gmp-1.0.0.0
recipModWord :: GmpLimb# -> GmpLimb# -> GmpLimb# Source
Version of recipModInteger
operating on Word#
s
Since: integer-gmp-1.0.0.0
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, or0#
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: integer-gmp-0.5.1.0
testPrimeBigNat :: BigNat -> Int# -> Int# Source
Version of testPrimeInteger
operating on BigNat
s
Since: integer-gmp-1.0.0.0
testPrimeWord# :: GmpLimb# -> Int# -> Int# Source
Version of testPrimeInteger
operating on Word#
s
Since: integer-gmp-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: integer-gmp-0.5.1.0
nextPrimeBigNat :: BigNat -> BigNat Source
Version of nextPrimeInteger
operating on BigNat
s
Since: integer-gmp-1.0.0.0
nextPrimeWord# :: GmpLimb# -> GmpLimb# Source
Version of nextPrimeInteger
operating on Word#
s
Since: integer-gmp-1.0.0.0
sizeInBaseBigNat :: BigNat -> Int# -> Word# Source
Version of sizeInBaseInteger
operating on BigNat
Since: integer-gmp-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
).base >= 2#
and base <= 256#
(Note: the documentation claims that only base <= 62#
is supported, however the actual implementation supports up to base 256).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: integer-gmp-0.5.1.0
sizeInBaseWord# :: Word# -> Int# -> Word# Source
Version of sizeInBaseInteger
operating on Word#
Since: integer-gmp-1.0.0.0
exportBigNatToAddr :: BigNat -> Addr# -> Int# -> IO Word Source
Version of exportIntegerToAddr
operating on BigNat
s.
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: integer-gmp-1.0.0.0
exportWordToAddr :: Word -> Addr# -> Int# -> IO Word Source
Version of exportIntegerToAddr
operating on Word
s.
exportBigNatToMutableByteArray :: BigNat -> MutableByteArray# RealWorld -> Word# -> Int# -> IO Word Source
Version of exportIntegerToMutableByteArray
operating on BigNat
s.
Since: integer-gmp-1.0.0.0
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
Integer
i
MutableByteArray#
mba
starting at offset
msbf
is 1#
or least significant byte first if msbf
is 0#
, andUse "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: integer-gmp-1.0.0.0
exportWordToMutableByteArray :: Word -> MutableByteArray# RealWorld -> Word# -> Int# -> IO Word Source
Version of exportIntegerToMutableByteArray
operating on Word
s.
Since: integer-gmp-1.0.0.0
importBigNatFromAddr :: Addr# -> Word# -> Int# -> IO BigNat Source
Version of importIntegerFromAddr
constructing a BigNat
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: integer-gmp-1.0.0.0
importBigNatFromByteArray :: ByteArray# -> Word# -> Word# -> Int# -> BigNat Source
Version of importIntegerFromByteArray
constructing a BigNat
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
msbf
is 1#
or least significant byte first if msbf
is 0#
, andInteger
Since: integer-gmp-1.0.0.0
© The University of Glasgow and others
Licensed under a BSD-style license (see top of the page).
https://downloads.haskell.org/~ghc/8.6.1/docs/html/libraries/integer-gmp-1.0.2.0/GHC-Integer-GMP-Internals.html