runtime/Lib/site-packages/Crypto/Math/_IntegerGMP.py

# ===================================================================
#
# Copyright (c) 2014, Legrandin <helderijs@gmail.com>
# All rights reserved.
#
# Redistribution and use in source and binary forms, with or without
# modification, are permitted provided that the following conditions
# are met:
#
# 1. Redistributions of source code must retain the above copyright
#    notice, this list of conditions and the following disclaimer.
# 2. Redistributions in binary form must reproduce the above copyright
#    notice, this list of conditions and the following disclaimer in
#    the documentation and/or other materials provided with the
#    distribution.
#
# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
# "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
# LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
# FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
# COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
# INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
# BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
# LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
# CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
# LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
# ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
# POSSIBILITY OF SUCH DAMAGE.
# ===================================================================

import sys

from Crypto.Util.py3compat import tobytes, is_native_int

from Crypto.Util._raw_api import (backend, load_lib,
                                  get_raw_buffer, get_c_string,
                                  null_pointer, create_string_buffer,
                                  c_ulong, c_size_t, c_uint8_ptr)

from ._IntegerBase import IntegerBase

gmp_defs = """typedef unsigned long UNIX_ULONG;
        typedef struct { int a; int b; void *c; } MPZ;
        typedef MPZ mpz_t[1];
        typedef UNIX_ULONG mp_bitcnt_t;

        void __gmpz_init (mpz_t x);
        void __gmpz_init_set (mpz_t rop, const mpz_t op);
        void __gmpz_init_set_ui (mpz_t rop, UNIX_ULONG op);

        UNIX_ULONG __gmpz_get_ui (const mpz_t op);
        void __gmpz_set (mpz_t rop, const mpz_t op);
        void __gmpz_set_ui (mpz_t rop, UNIX_ULONG op);
        void __gmpz_add (mpz_t rop, const mpz_t op1, const mpz_t op2);
        void __gmpz_add_ui (mpz_t rop, const mpz_t op1, UNIX_ULONG op2);
        void __gmpz_sub_ui (mpz_t rop, const mpz_t op1, UNIX_ULONG op2);
        void __gmpz_addmul (mpz_t rop, const mpz_t op1, const mpz_t op2);
        void __gmpz_addmul_ui (mpz_t rop, const mpz_t op1, UNIX_ULONG op2);
        void __gmpz_submul_ui (mpz_t rop, const mpz_t op1, UNIX_ULONG op2);
        void __gmpz_import (mpz_t rop, size_t count, int order, size_t size,
                            int endian, size_t nails, const void *op);
        void * __gmpz_export (void *rop, size_t *countp, int order,
                              size_t size,
                              int endian, size_t nails, const mpz_t op);
        size_t __gmpz_sizeinbase (const mpz_t op, int base);
        void __gmpz_sub (mpz_t rop, const mpz_t op1, const mpz_t op2);
        void __gmpz_mul (mpz_t rop, const mpz_t op1, const mpz_t op2);
        void __gmpz_mul_ui (mpz_t rop, const mpz_t op1, UNIX_ULONG op2);
        int __gmpz_cmp (const mpz_t op1, const mpz_t op2);
        void __gmpz_powm (mpz_t rop, const mpz_t base, const mpz_t exp, const
                          mpz_t mod);
        void __gmpz_powm_ui (mpz_t rop, const mpz_t base, UNIX_ULONG exp,
                             const mpz_t mod);
        void __gmpz_pow_ui (mpz_t rop, const mpz_t base, UNIX_ULONG exp);
        void __gmpz_sqrt(mpz_t rop, const mpz_t op);
        void __gmpz_mod (mpz_t r, const mpz_t n, const mpz_t d);
        void __gmpz_neg (mpz_t rop, const mpz_t op);
        void __gmpz_abs (mpz_t rop, const mpz_t op);
        void __gmpz_and (mpz_t rop, const mpz_t op1, const mpz_t op2);
        void __gmpz_ior (mpz_t rop, const mpz_t op1, const mpz_t op2);
        void __gmpz_clear (mpz_t x);
        void __gmpz_tdiv_q_2exp (mpz_t q, const mpz_t n, mp_bitcnt_t b);
        void __gmpz_fdiv_q (mpz_t q, const mpz_t n, const mpz_t d);
        void __gmpz_mul_2exp (mpz_t rop, const mpz_t op1, mp_bitcnt_t op2);
        int __gmpz_tstbit (const mpz_t op, mp_bitcnt_t bit_index);
        int __gmpz_perfect_square_p (const mpz_t op);
        int __gmpz_jacobi (const mpz_t a, const mpz_t b);
        void __gmpz_gcd (mpz_t rop, const mpz_t op1, const mpz_t op2);
        UNIX_ULONG __gmpz_gcd_ui (mpz_t rop, const mpz_t op1,
                                     UNIX_ULONG op2);
        void __gmpz_lcm (mpz_t rop, const mpz_t op1, const mpz_t op2);
        int __gmpz_invert (mpz_t rop, const mpz_t op1, const mpz_t op2);
        int __gmpz_divisible_p (const mpz_t n, const mpz_t d);
        int __gmpz_divisible_ui_p (const mpz_t n, UNIX_ULONG d);
        """

if sys.platform == "win32":
    raise ImportError("Not using GMP on Windows")

lib = load_lib("gmp", gmp_defs)
implementation = {"library": "gmp", "api": backend}

if hasattr(lib, "__mpir_version"):
    raise ImportError("MPIR library detected")

# In order to create a function that returns a pointer to
# a new MPZ structure, we need to break the abstraction
# and know exactly what ffi backend we have
if implementation["api"] == "ctypes":
    from ctypes import Structure, c_int, c_void_p, byref

    class _MPZ(Structure):
        _fields_ = [('_mp_alloc', c_int),
                    ('_mp_size', c_int),
                    ('_mp_d', c_void_p)]

    def new_mpz():
        return byref(_MPZ())

else:
    # We are using CFFI
    from Crypto.Util._raw_api import ffi

    def new_mpz():
        return ffi.new("MPZ*")


# Lazy creation of GMP methods
class _GMP(object):

    def __getattr__(self, name):
        if name.startswith("mpz_"):
            func_name = "__gmpz_" + name[4:]
        elif name.startswith("gmp_"):
            func_name = "__gmp_" + name[4:]
        else:
            raise AttributeError("Attribute %s is invalid" % name)
        func = getattr(lib, func_name)
        setattr(self, name, func)
        return func


_gmp = _GMP()


class IntegerGMP(IntegerBase):
    """A fast, arbitrary precision integer"""

    _zero_mpz_p = new_mpz()
    _gmp.mpz_init_set_ui(_zero_mpz_p, c_ulong(0))

    def __init__(self, value):
        """Initialize the integer to the given value."""

        self._mpz_p = new_mpz()
        self._initialized = False

        if isinstance(value, float):
            raise ValueError("A floating point type is not a natural number")

        if is_native_int(value):
            _gmp.mpz_init(self._mpz_p)
            self._initialized = True
            if value == 0:
                return

            tmp = new_mpz()
            _gmp.mpz_init(tmp)

            try:
                positive = value >= 0
                reduce = abs(value)
                slots = (reduce.bit_length() - 1) // 32 + 1

                while slots > 0:
                    slots = slots - 1
                    _gmp.mpz_set_ui(tmp,
                                    c_ulong(0xFFFFFFFF & (reduce >> (slots * 32))))
                    _gmp.mpz_mul_2exp(tmp, tmp, c_ulong(slots * 32))
                    _gmp.mpz_add(self._mpz_p, self._mpz_p, tmp)
            finally:
                _gmp.mpz_clear(tmp)

            if not positive:
                _gmp.mpz_neg(self._mpz_p, self._mpz_p)

        elif isinstance(value, IntegerGMP):
            _gmp.mpz_init_set(self._mpz_p, value._mpz_p)
            self._initialized = True
        else:
            raise NotImplementedError


    # Conversions
    def __int__(self):
        tmp = new_mpz()
        _gmp.mpz_init_set(tmp, self._mpz_p)

        try:
            value = 0
            slot = 0
            while _gmp.mpz_cmp(tmp, self._zero_mpz_p) != 0:
                lsb = _gmp.mpz_get_ui(tmp) & 0xFFFFFFFF
                value |= lsb << (slot * 32)
                _gmp.mpz_tdiv_q_2exp(tmp, tmp, c_ulong(32))
                slot = slot + 1
        finally:
            _gmp.mpz_clear(tmp)

        if self < 0:
            value = -value
        return int(value)

    def __str__(self):
        return str(int(self))

    def __repr__(self):
        return "Integer(%s)" % str(self)

    # Only Python 2.x
    def __hex__(self):
        return hex(int(self))

    # Only Python 3.x
    def __index__(self):
        return int(self)

    def to_bytes(self, block_size=0, byteorder='big'):
        """Convert the number into a byte string.

        This method encodes the number in network order and prepends
        as many zero bytes as required. It only works for non-negative
        values.

        :Parameters:
          block_size : integer
            The exact size the output byte string must have.
            If zero, the string has the minimal length.
          byteorder : string
            'big' for big-endian integers (default), 'little' for litte-endian.
        :Returns:
          A byte string.
        :Raise ValueError:
          If the value is negative or if ``block_size`` is
          provided and the length of the byte string would exceed it.
        """

        if self < 0:
            raise ValueError("Conversion only valid for non-negative numbers")

        buf_len = (_gmp.mpz_sizeinbase(self._mpz_p, 2) + 7) // 8
        if buf_len > block_size > 0:
            raise ValueError("Number is too big to convert to byte string"
                             " of prescribed length")
        buf = create_string_buffer(buf_len)


        _gmp.mpz_export(
                buf,
                null_pointer,  # Ignore countp
                1,             # Big endian
                c_size_t(1),   # Each word is 1 byte long
                0,             # Endianess within a word - not relevant
                c_size_t(0),   # No nails
                self._mpz_p)

        result = b'\x00' * max(0, block_size - buf_len) + get_raw_buffer(buf)
        if byteorder == 'big':
            pass
        elif byteorder == 'little':
            result = bytearray(result)
            result.reverse()
            result = bytes(result)
        else:
            raise ValueError("Incorrect byteorder")
        return result

    @staticmethod
    def from_bytes(byte_string, byteorder='big'):
        """Convert a byte string into a number.

        :Parameters:
          byte_string : byte string
            The input number, encoded in network order.
            It can only be non-negative.
          byteorder : string
            'big' for big-endian integers (default), 'little' for litte-endian.

        :Return:
          The ``Integer`` object carrying the same value as the input.
        """
        result = IntegerGMP(0)
        if byteorder == 'big':
            pass
        elif byteorder == 'little':
            byte_string = bytearray(byte_string)
            byte_string.reverse()
        else:
            raise ValueError("Incorrect byteorder")
        _gmp.mpz_import(
                        result._mpz_p,
                        c_size_t(len(byte_string)),  # Amount of words to read
                        1,            # Big endian
                        c_size_t(1),  # Each word is 1 byte long
                        0,            # Endianess within a word - not relevant
                        c_size_t(0),  # No nails
                        c_uint8_ptr(byte_string))
        return result

    # Relations
    def _apply_and_return(self, func, term):
        if not isinstance(term, IntegerGMP):
            term = IntegerGMP(term)
        return func(self._mpz_p, term._mpz_p)

    def __eq__(self, term):
        if not (isinstance(term, IntegerGMP) or is_native_int(term)):
            return False
        return self._apply_and_return(_gmp.mpz_cmp, term) == 0

    def __ne__(self, term):
        if not (isinstance(term, IntegerGMP) or is_native_int(term)):
            return True
        return self._apply_and_return(_gmp.mpz_cmp, term) != 0

    def __lt__(self, term):
        return self._apply_and_return(_gmp.mpz_cmp, term) < 0

    def __le__(self, term):
        return self._apply_and_return(_gmp.mpz_cmp, term) <= 0

    def __gt__(self, term):
        return self._apply_and_return(_gmp.mpz_cmp, term) > 0

    def __ge__(self, term):
        return self._apply_and_return(_gmp.mpz_cmp, term) >= 0

    def __nonzero__(self):
        return _gmp.mpz_cmp(self._mpz_p, self._zero_mpz_p) != 0
    __bool__ = __nonzero__

    def is_negative(self):
        return _gmp.mpz_cmp(self._mpz_p, self._zero_mpz_p) < 0

    # Arithmetic operations
    def __add__(self, term):
        result = IntegerGMP(0)
        if not isinstance(term, IntegerGMP):
            try:
                term = IntegerGMP(term)
            except NotImplementedError:
                return NotImplemented
        _gmp.mpz_add(result._mpz_p,
                     self._mpz_p,
                     term._mpz_p)
        return result

    def __sub__(self, term):
        result = IntegerGMP(0)
        if not isinstance(term, IntegerGMP):
            try:
                term = IntegerGMP(term)
            except NotImplementedError:
                return NotImplemented
        _gmp.mpz_sub(result._mpz_p,
                     self._mpz_p,
                     term._mpz_p)
        return result

    def __mul__(self, term):
        result = IntegerGMP(0)
        if not isinstance(term, IntegerGMP):
            try:
                term = IntegerGMP(term)
            except NotImplementedError:
                return NotImplemented
        _gmp.mpz_mul(result._mpz_p,
                     self._mpz_p,
                     term._mpz_p)
        return result

    def __floordiv__(self, divisor):
        if not isinstance(divisor, IntegerGMP):
            divisor = IntegerGMP(divisor)
        if _gmp.mpz_cmp(divisor._mpz_p,
                        self._zero_mpz_p) == 0:
            raise ZeroDivisionError("Division by zero")
        result = IntegerGMP(0)
        _gmp.mpz_fdiv_q(result._mpz_p,
                        self._mpz_p,
                        divisor._mpz_p)
        return result

    def __mod__(self, divisor):
        if not isinstance(divisor, IntegerGMP):
            divisor = IntegerGMP(divisor)
        comp = _gmp.mpz_cmp(divisor._mpz_p,
                            self._zero_mpz_p)
        if comp == 0:
            raise ZeroDivisionError("Division by zero")
        if comp < 0:
            raise ValueError("Modulus must be positive")
        result = IntegerGMP(0)
        _gmp.mpz_mod(result._mpz_p,
                     self._mpz_p,
                     divisor._mpz_p)
        return result

    def inplace_pow(self, exponent, modulus=None):

        if modulus is None:
            if exponent < 0:
                raise ValueError("Exponent must not be negative")

            # Normal exponentiation
            if exponent > 256:
                raise ValueError("Exponent is too big")
            _gmp.mpz_pow_ui(self._mpz_p,
                            self._mpz_p,   # Base
                            c_ulong(int(exponent))
                            )
        else:
            # Modular exponentiation
            if not isinstance(modulus, IntegerGMP):
                modulus = IntegerGMP(modulus)
            if not modulus:
                raise ZeroDivisionError("Division by zero")
            if modulus.is_negative():
                raise ValueError("Modulus must be positive")
            if is_native_int(exponent):
                if exponent < 0:
                    raise ValueError("Exponent must not be negative")
                if exponent < 65536:
                    _gmp.mpz_powm_ui(self._mpz_p,
                                     self._mpz_p,
                                     c_ulong(exponent),
                                     modulus._mpz_p)
                    return self
                exponent = IntegerGMP(exponent)
            elif exponent.is_negative():
                raise ValueError("Exponent must not be negative")
            _gmp.mpz_powm(self._mpz_p,
                          self._mpz_p,
                          exponent._mpz_p,
                          modulus._mpz_p)
        return self

    def __pow__(self, exponent, modulus=None):
        result = IntegerGMP(self)
        return result.inplace_pow(exponent, modulus)

    def __abs__(self):
        result = IntegerGMP(0)
        _gmp.mpz_abs(result._mpz_p, self._mpz_p)
        return result

    def sqrt(self, modulus=None):
        """Return the largest Integer that does not
        exceed the square root"""

        if modulus is None:
            if self < 0:
                raise ValueError("Square root of negative value")
            result = IntegerGMP(0)
            _gmp.mpz_sqrt(result._mpz_p,
                          self._mpz_p)
        else:
            if modulus <= 0:
                raise ValueError("Modulus must be positive")
            modulus = int(modulus)
            result = IntegerGMP(self._tonelli_shanks(int(self) % modulus, modulus))

        return result

    def __iadd__(self, term):
        if is_native_int(term):
            if 0 <= term < 65536:
                _gmp.mpz_add_ui(self._mpz_p,
                                self._mpz_p,
                                c_ulong(term))
                return self
            if -65535 < term < 0:
                _gmp.mpz_sub_ui(self._mpz_p,
                                self._mpz_p,
                                c_ulong(-term))
                return self
            term = IntegerGMP(term)
        _gmp.mpz_add(self._mpz_p,
                     self._mpz_p,
                     term._mpz_p)
        return self

    def __isub__(self, term):
        if is_native_int(term):
            if 0 <= term < 65536:
                _gmp.mpz_sub_ui(self._mpz_p,
                                self._mpz_p,
                                c_ulong(term))
                return self
            if -65535 < term < 0:
                _gmp.mpz_add_ui(self._mpz_p,
                                self._mpz_p,
                                c_ulong(-term))
                return self
            term = IntegerGMP(term)
        _gmp.mpz_sub(self._mpz_p,
                     self._mpz_p,
                     term._mpz_p)
        return self

    def __imul__(self, term):
        if is_native_int(term):
            if 0 <= term < 65536:
                _gmp.mpz_mul_ui(self._mpz_p,
                                self._mpz_p,
                                c_ulong(term))
                return self
            if -65535 < term < 0:
                _gmp.mpz_mul_ui(self._mpz_p,
                                self._mpz_p,
                                c_ulong(-term))
                _gmp.mpz_neg(self._mpz_p, self._mpz_p)
                return self
            term = IntegerGMP(term)
        _gmp.mpz_mul(self._mpz_p,
                     self._mpz_p,
                     term._mpz_p)
        return self

    def __imod__(self, divisor):
        if not isinstance(divisor, IntegerGMP):
            divisor = IntegerGMP(divisor)
        comp = _gmp.mpz_cmp(divisor._mpz_p,
                            divisor._zero_mpz_p)
        if comp == 0:
            raise ZeroDivisionError("Division by zero")
        if comp < 0:
            raise ValueError("Modulus must be positive")
        _gmp.mpz_mod(self._mpz_p,
                     self._mpz_p,
                     divisor._mpz_p)
        return self

    # Boolean/bit operations
    def __and__(self, term):
        result = IntegerGMP(0)
        if not isinstance(term, IntegerGMP):
            term = IntegerGMP(term)
        _gmp.mpz_and(result._mpz_p,
                     self._mpz_p,
                     term._mpz_p)
        return result

    def __or__(self, term):
        result = IntegerGMP(0)
        if not isinstance(term, IntegerGMP):
            term = IntegerGMP(term)
        _gmp.mpz_ior(result._mpz_p,
                     self._mpz_p,
                     term._mpz_p)
        return result

    def __rshift__(self, pos):
        result = IntegerGMP(0)
        if pos < 0:
            raise ValueError("negative shift count")
        if pos > 65536:
            if self < 0:
                return -1
            else:
                return 0
        _gmp.mpz_tdiv_q_2exp(result._mpz_p,
                             self._mpz_p,
                             c_ulong(int(pos)))
        return result

    def __irshift__(self, pos):
        if pos < 0:
            raise ValueError("negative shift count")
        if pos > 65536:
            if self < 0:
                return -1
            else:
                return 0
        _gmp.mpz_tdiv_q_2exp(self._mpz_p,
                             self._mpz_p,
                             c_ulong(int(pos)))
        return self

    def __lshift__(self, pos):
        result = IntegerGMP(0)
        if not 0 <= pos < 65536:
            raise ValueError("Incorrect shift count")
        _gmp.mpz_mul_2exp(result._mpz_p,
                          self._mpz_p,
                          c_ulong(int(pos)))
        return result

    def __ilshift__(self, pos):
        if not 0 <= pos < 65536:
            raise ValueError("Incorrect shift count")
        _gmp.mpz_mul_2exp(self._mpz_p,
                          self._mpz_p,
                          c_ulong(int(pos)))
        return self

    def get_bit(self, n):
        """Return True if the n-th bit is set to 1.
        Bit 0 is the least significant."""

        if self < 0:
            raise ValueError("no bit representation for negative values")
        if n < 0:
            raise ValueError("negative bit count")
        if n > 65536:
            return 0
        return bool(_gmp.mpz_tstbit(self._mpz_p,
                                    c_ulong(int(n))))

    # Extra
    def is_odd(self):
        return _gmp.mpz_tstbit(self._mpz_p, 0) == 1

    def is_even(self):
        return _gmp.mpz_tstbit(self._mpz_p, 0) == 0

    def size_in_bits(self):
        """Return the minimum number of bits that can encode the number."""

        if self < 0:
            raise ValueError("Conversion only valid for non-negative numbers")
        return _gmp.mpz_sizeinbase(self._mpz_p, 2)

    def size_in_bytes(self):
        """Return the minimum number of bytes that can encode the number."""
        return (self.size_in_bits() - 1) // 8 + 1

    def is_perfect_square(self):
        return _gmp.mpz_perfect_square_p(self._mpz_p) != 0

    def fail_if_divisible_by(self, small_prime):
        """Raise an exception if the small prime is a divisor."""

        if is_native_int(small_prime):
            if 0 < small_prime < 65536:
                if _gmp.mpz_divisible_ui_p(self._mpz_p,
                                           c_ulong(small_prime)):
                    raise ValueError("The value is composite")
                return
            small_prime = IntegerGMP(small_prime)
        if _gmp.mpz_divisible_p(self._mpz_p,
                                small_prime._mpz_p):
            raise ValueError("The value is composite")

    def multiply_accumulate(self, a, b):
        """Increment the number by the product of a and b."""

        if not isinstance(a, IntegerGMP):
            a = IntegerGMP(a)
        if is_native_int(b):
            if 0 < b < 65536:
                _gmp.mpz_addmul_ui(self._mpz_p,
                                   a._mpz_p,
                                   c_ulong(b))
                return self
            if -65535 < b < 0:
                _gmp.mpz_submul_ui(self._mpz_p,
                                   a._mpz_p,
                                   c_ulong(-b))
                return self
            b = IntegerGMP(b)
        _gmp.mpz_addmul(self._mpz_p,
                        a._mpz_p,
                        b._mpz_p)
        return self

    def set(self, source):
        """Set the Integer to have the given value"""

        if not isinstance(source, IntegerGMP):
            source = IntegerGMP(source)
        _gmp.mpz_set(self._mpz_p,
                     source._mpz_p)
        return self

    def inplace_inverse(self, modulus):
        """Compute the inverse of this number in the ring of
        modulo integers.

        Raise an exception if no inverse exists.
        """

        if not isinstance(modulus, IntegerGMP):
            modulus = IntegerGMP(modulus)

        comp = _gmp.mpz_cmp(modulus._mpz_p,
                            self._zero_mpz_p)
        if comp == 0:
            raise ZeroDivisionError("Modulus cannot be zero")
        if comp < 0:
            raise ValueError("Modulus must be positive")

        result = _gmp.mpz_invert(self._mpz_p,
                                 self._mpz_p,
                                 modulus._mpz_p)
        if not result:
            raise ValueError("No inverse value can be computed")
        return self

    def inverse(self, modulus):
        result = IntegerGMP(self)
        result.inplace_inverse(modulus)
        return result

    def gcd(self, term):
        """Compute the greatest common denominator between this
        number and another term."""

        result = IntegerGMP(0)
        if is_native_int(term):
            if 0 < term < 65535:
                _gmp.mpz_gcd_ui(result._mpz_p,
                                self._mpz_p,
                                c_ulong(term))
                return result
            term = IntegerGMP(term)
        _gmp.mpz_gcd(result._mpz_p, self._mpz_p, term._mpz_p)
        return result

    def lcm(self, term):
        """Compute the least common multiplier between this
        number and another term."""

        result = IntegerGMP(0)
        if not isinstance(term, IntegerGMP):
            term = IntegerGMP(term)
        _gmp.mpz_lcm(result._mpz_p, self._mpz_p, term._mpz_p)
        return result

    @staticmethod
    def jacobi_symbol(a, n):
        """Compute the Jacobi symbol"""

        if not isinstance(a, IntegerGMP):
            a = IntegerGMP(a)
        if not isinstance(n, IntegerGMP):
            n = IntegerGMP(n)
        if n <= 0 or n.is_even():
            raise ValueError("n must be positive odd for the Jacobi symbol")
        return _gmp.mpz_jacobi(a._mpz_p, n._mpz_p)

    # Clean-up
    def __del__(self):

        try:
            if self._mpz_p is not None:
                if self._initialized:
                    _gmp.mpz_clear(self._mpz_p)

            self._mpz_p = None
        except AttributeError:
            pass