r"""
Ring `\ZZ/n\ZZ` of integers modulo `n`

EXAMPLES::

    sage: R = Integers(97)
    sage: a = R(5)
    sage: a**100000000000000000000000000000000000000000000000000000000000000
    61

This example illustrates the relation between
`\ZZ/p\ZZ` and `\GF{p}`. In
particular, there is a canonical map to `\GF{p}`, but not in
the other direction.

::

    sage: r = Integers(7)
    sage: s = GF(7)
    sage: r.has_coerce_map_from(s)
    False
    sage: s.has_coerce_map_from(r)
    True
    sage: s(1) + r(1)
    2
    sage: parent(s(1) + r(1))
    Finite Field of size 7
    sage: parent(r(1) + s(1))
    Finite Field of size 7

We list the elements of `\ZZ/3\ZZ`::

    sage: R = Integers(3)
    sage: list(R)
    [0, 1, 2]

AUTHORS:

- William Stein (initial code)

- David Joyner (2005-12-22): most examples

- Robert Bradshaw (2006-08-24): convert to SageX (Cython)

- William Stein (2007-04-29): square_roots_of_one

- Simon King (2011-04-21): allow to prescribe a category

- Simon King (2013-09): Only allow to prescribe the category of fields
"""

#*****************************************************************************
#       Copyright (C) 2005 William Stein <wstein@gmail.com>
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 2 of the License, or
# (at your option) any later version.
#                  http://www.gnu.org/licenses/
#*****************************************************************************

from __future__ import absolute_import, print_function

import sage.misc.prandom as random

from sage.arith.all import factor, primitive_root, CRT_basis
import sage.rings.ring as ring
from . import integer_mod
import sage.rings.integer as integer
import sage.rings.integer_ring as integer_ring
import sage.rings.quotient_ring as quotient_ring

from sage.libs.pari.all import pari, PariError

import sage.interfaces.all
from sage.misc.cachefunc import cached_method

from sage.structure.factory import UniqueFactory

class IntegerModFactory(UniqueFactory):
    r"""
    Return the quotient ring `\ZZ / n\ZZ`.

    INPUT:

    - ``order`` -- integer (default: 0); positive or negative
    - ``is_field`` -- bool (default: ``False``); assert that
      the order is prime and hence the quotient ring belongs to
      the category of fields

    .. NOTE::

        The optional argument ``is_field`` is not part of the cache key.
        Hence, this factory will create precisely one instance of `\ZZ /
        n\ZZ`.  However, if ``is_field`` is true, then a previously created
        instance of the quotient ring will be updated to be in the category of
        fields.

        **Use with care!** Erroneously putting `\ZZ / n\ZZ` into the category
        of fields may have consequences that can compromise a whole Sage
        session, so that a restart will be needed.

    EXAMPLES::

        sage: IntegerModRing(15)
        Ring of integers modulo 15
        sage: IntegerModRing(7)
        Ring of integers modulo 7
        sage: IntegerModRing(-100)
        Ring of integers modulo 100

    Note that you can also use ``Integers``, which is a
    synonym for ``IntegerModRing``.

    ::

        sage: Integers(18)
        Ring of integers modulo 18
        sage: Integers() is Integers(0) is ZZ
        True

    .. NOTE::

        Testing whether a quotient ring `\ZZ / n\ZZ` is a field can of
        course be very costly. By default, it is not tested whether `n`
        is prime or not, in contrast to
        :func:`~sage.rings.finite_rings.finite_field_constructor.GF`. If the user
        is sure that the modulus is prime and wants to avoid a primality
        test, (s)he can provide ``category=Fields()`` when constructing
        the quotient ring, and then the result will behave like a field.
        If the category is not provided during initialisation, and it is
        found out later that the ring is in fact a field, then the category
        will be changed at runtime, having the same effect as providing
        ``Fields()`` during initialisation.

    EXAMPLES::

        sage: R = IntegerModRing(5)
        sage: R.category()
        Join of Category of finite commutative rings
            and Category of subquotients of monoids
            and Category of quotients of semigroups
            and Category of finite enumerated sets
        sage: R in Fields()
        True
        sage: R.category()
        Join of Category of finite fields
            and Category of subquotients of monoids
            and Category of quotients of semigroups
        sage: S = IntegerModRing(5, is_field=True)
        sage: S is R
        True

    .. WARNING::

        If the optional argument ``is_field`` was used by mistake, there is
        currently no way to revert its impact, even though
        :meth:`IntegerModRing_generic.is_field` with the optional argument
        ``proof=True`` would return the correct answer.  So, prescribe
        ``is_field=True`` only if you know what your are doing!

    EXAMPLES::

        sage: R = IntegerModRing(33, is_field=True)
        sage: R in Fields()
        True
        sage: R.is_field()
        True

    If the optional argument `proof=True` is provided, primality is tested and
    the mistaken category assignment is reported::

        sage: R.is_field(proof=True)
        Traceback (most recent call last):
        ...
        ValueError: THIS SAGE SESSION MIGHT BE SERIOUSLY COMPROMISED!
        The order 33 is not prime, but this ring has been put
        into the category of fields. This may already have consequences
        in other parts of Sage. Either it was a mistake of the user,
        or a probabilistic primality test has failed.
        In the latter case, please inform the developers.

    However, the mistaken assignment is not automatically corrected::

        sage: R in Fields()
        True

    """
    def get_object(self, version, key, extra_args):
        out = super(IntegerModFactory,self).get_object(version, key, extra_args)
        category = extra_args.get('category', None)
        if category is not None:
            out._refine_category_(category)
            out._factory_data[3]['category'] = category
        return out

    def create_key_and_extra_args(self, order=0, is_field=False):
        """
        An integer mod ring is specified uniquely by its order.

        EXAMPLES::

            sage: Zmod.create_key_and_extra_args(7)
            (7, {})
            sage: Zmod.create_key_and_extra_args(7, True)
            (7, {'category': Category of fields})
        """
        if is_field:
            from sage.categories.fields import Fields
            return order, {'category':Fields()}
        return order, {}

    def create_object(self, version, order, **kwds):
        """
        EXAMPLES::

            sage: R = Integers(10)
            sage: TestSuite(R).run() # indirect doctest
        """
        if isinstance(order, tuple):
            # this is for unpickling old data
            order, category = order
            kwds.setdefault('category', category)
        if order < 0:
            order = -order
        if order == 0:
            return integer_ring.IntegerRing(**kwds)
        else:
            return IntegerModRing_generic(order, **kwds)

Zmod = Integers = IntegerModRing = IntegerModFactory("IntegerModRing")


def is_IntegerModRing(x):
    """
    Return ``True`` if ``x`` is an integer modulo ring.

    EXAMPLES::

        sage: from sage.rings.finite_rings.integer_mod_ring import is_IntegerModRing
        sage: R = IntegerModRing(17)
        sage: is_IntegerModRing(R)
        True
        sage: is_IntegerModRing(GF(13))
        True
        sage: is_IntegerModRing(GF(4, 'a'))
        False
        sage: is_IntegerModRing(10)
        False
        sage: is_IntegerModRing(ZZ)
        False
    """
    return isinstance(x, IntegerModRing_generic)

from sage.categories.commutative_rings import CommutativeRings
from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
from sage.categories.category import JoinCategory
default_category = JoinCategory((CommutativeRings(), FiniteEnumeratedSets()))
ZZ = integer_ring.IntegerRing()


def _unit_gens_primepowercase(p, r):
    r"""
    Return a list of generators for `(\ZZ/p^r\ZZ)^*` and their orders.

    EXAMPLES::

        sage: from sage.rings.finite_rings.integer_mod_ring import _unit_gens_primepowercase
        sage: _unit_gens_primepowercase(2, 3)
        [(7, 2), (5, 2)]
        sage: _unit_gens_primepowercase(17, 1)
        [(3, 16)]
        sage: _unit_gens_primepowercase(3, 3)
        [(2, 18)]
    """
    pr = p**r
    if p == 2:
        if r == 1:
            return []
        if r == 2:
            return [(integer_mod.Mod(3, 4), integer.Integer(2))]
        return [(integer_mod.Mod(-1, pr), integer.Integer(2)),
                (integer_mod.Mod(5, pr), integer.Integer(2**(r - 2)))]

    # odd prime
    return [(integer_mod.Mod(primitive_root(pr, check=False), pr),
             integer.Integer(p**(r - 1) * (p - 1)))]


class IntegerModRing_generic(quotient_ring.QuotientRing_generic):
    """
    The ring of integers modulo `N`.

    INPUT:

    - ``order`` -- an integer

    - ``category`` -- a subcategory of ``CommutativeRings()`` (the default)

    OUTPUT:

    The ring of integers modulo `N`.

    EXAMPLES:

    First we compute with integers modulo `29`.

    ::

        sage: FF = IntegerModRing(29)
        sage: FF
        Ring of integers modulo 29
        sage: FF.category()
        Join of Category of finite commutative rings
            and Category of subquotients of monoids
            and Category of quotients of semigroups
            and Category of finite enumerated sets
        sage: FF.is_field()
        True
        sage: FF.characteristic()
        29
        sage: FF.order()
        29
        sage: gens = FF.unit_gens()
        sage: a = gens[0]
        sage: a
        2
        sage: a.is_square()
        False
        sage: def pow(i): return a**i
        sage: [pow(i) for i in range(16)]
        [1, 2, 4, 8, 16, 3, 6, 12, 24, 19, 9, 18, 7, 14, 28, 27]
        sage: TestSuite(FF).run()

    We have seen above that an integer mod ring is, by default, not
    initialised as an object in the category of fields. However, one
    can force it to be. Moreover, testing containment in the category
    of fields my re-initialise the category of the integer mod ring::

        sage: F19 = IntegerModRing(19, is_field=True)
        sage: F19.category().is_subcategory(Fields())
        True
        sage: F23 = IntegerModRing(23)
        sage: F23.category().is_subcategory(Fields())
        False
        sage: F23 in Fields()
        True
        sage: F23.category().is_subcategory(Fields())
        True
        sage: TestSuite(F19).run()
        sage: TestSuite(F23).run()

    By :trac:`15229`, there is a unique instance of the
    integral quotient ring of a given order. Using the
    :func:`IntegerModRing` factory twice, and using
    ``is_field=True`` the second time, will update the
    category of the unique instance::

        sage: F31a = IntegerModRing(31)
        sage: F31a.category().is_subcategory(Fields())
        False
        sage: F31b = IntegerModRing(31, is_field=True)
        sage: F31a is F31b
        True
        sage: F31a.category().is_subcategory(Fields())
        True

    Next we compute with the integers modulo `16`.

    ::

        sage: Z16 = IntegerModRing(16)
        sage: Z16.category()
        Join of Category of finite commutative rings
            and Category of subquotients of monoids
            and Category of quotients of semigroups
            and Category of finite enumerated sets
        sage: Z16.is_field()
        False
        sage: Z16.order()
        16
        sage: Z16.characteristic()
        16
        sage: gens = Z16.unit_gens()
        sage: gens
        (15, 5)
        sage: a = gens[0]
        sage: b = gens[1]
        sage: def powa(i): return a**i
        sage: def powb(i): return b**i
        sage: gp_exp = FF.unit_group_exponent()
        sage: gp_exp
        28
        sage: [powa(i) for i in range(15)]
        [1, 15, 1, 15, 1, 15, 1, 15, 1, 15, 1, 15, 1, 15, 1]
        sage: [powb(i) for i in range(15)]
        [1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9]
        sage: a.multiplicative_order()
        2
        sage: b.multiplicative_order()
        4
        sage: TestSuite(Z16).run()

    Saving and loading::

        sage: R = Integers(100000)
        sage: TestSuite(R).run()  # long time (17s on sage.math, 2011)

    Testing ideals and quotients::

        sage: Z10 = Integers(10)
        sage: I = Z10.principal_ideal(0)
        sage: Z10.quotient(I) == Z10
        True
        sage: I = Z10.principal_ideal(2)
        sage: Z10.quotient(I) == Z10
        False
        sage: I.is_prime()
        True

    ::

        sage: R = IntegerModRing(97)
        sage: a = R(5)
        sage: a**(10^62)
        61
    """
    def __init__(self, order, cache=None, category=None):
        """
        Create with the command ``IntegerModRing(order)``.

        TESTS::

            sage: FF = IntegerModRing(29)
            sage: TestSuite(FF).run()
            sage: F19 = IntegerModRing(19, is_field=True)
            sage: TestSuite(F19).run()
            sage: F23 = IntegerModRing(23)
            sage: F23 in Fields()
            True
            sage: TestSuite(F23).run()
            sage: Z16 = IntegerModRing(16)
            sage: TestSuite(Z16).run()
            sage: R = Integers(100000)
            sage: TestSuite(R).run()  # long time (17s on sage.math, 2011)
        """
        order = ZZ(order)
        if order <= 0:
            raise ZeroDivisionError("order must be positive")
        self.__order = order
        self._pyx_order = integer_mod.NativeIntStruct(order)
        global default_category
        if category is None:
            category = default_category
        else:
            # If the category is given, e.g., as Fields(), then we still
            # know that the result will also live in default_category.
            # Hence, we use the join of the default and the given category.
            category = category.join([category,default_category])
        # Give the generator a 'name' to make quotients work.  The
        # name 'x' is used because it's also used for the ring of
        # integers: see the __init__ method for IntegerRing_class in
        # sage/rings/integer_ring.pyx.
        quotient_ring.QuotientRing_generic.__init__(self, ZZ, ZZ.ideal(order),
                                                    names=('x',),
                                                    category=category)
        # We want that the ring is its own base ring.
        self._base = self
        if cache is None:
            cache = order < 500
        if cache:
            self._precompute_table()
        self._zero_element = integer_mod.IntegerMod(self, 0)
        self._one_element = integer_mod.IntegerMod(self, 1)

    def _macaulay2_init_(self):
        """
        EXAMPLES::

            sage: macaulay2(Integers(7))  # optional - macaulay2
            ZZ
            --
             7

        ::

            sage: macaulay2(Integers(10)) # optional - macaulay2
            Traceback (most recent call last):
            ...
            TypeError: Error evaluating Macaulay2 code.
            IN:sage1=ZZ/10;
            OUT:...error: ZZ/n not implemented yet for composite n
        """
        return "ZZ/{}".format(self.order())

    def _axiom_init_(self):
        """
        Returns a string representation of self in (Pan)Axiom.

        EXAMPLES::

            sage: Z7 = Integers(7)
            sage: Z7._axiom_init_()
            'IntegerMod(7)'

            sage: axiom(Z7)  #optional - axiom
            IntegerMod 7

            sage: fricas(Z7) #optional - fricas
            IntegerMod(7)
        """
        return 'IntegerMod({})'.format(self.order())

    _fricas_init_ = _axiom_init_

    def krull_dimension(self):
        """
        Return the Krull dimension of ``self``.

        EXAMPLES::

            sage: Integers(18).krull_dimension()
            0
        """
        return integer.Integer(0)

    def is_noetherian(self):
        """
        Check if ``self`` is a Noetherian ring.

        EXAMPLES::

            sage: Integers(8).is_noetherian()
            True
        """
        return True

    def extension(self, poly, name=None, names=None, embedding=None):
        """
        Return an algebraic extension of ``self``. See
        :meth:`sage.rings.ring.CommutativeRing.extension()` for more
        information.

        EXAMPLES::

            sage: R.<t> = QQ[]
            sage: Integers(8).extension(t^2 - 3)
            Univariate Quotient Polynomial Ring in t over Ring of integers modulo 8 with modulus t^2 + 5
        """
        if self.modulus() == 1:
            return self

        from sage.rings.ring import CommutativeRing
        return CommutativeRing.extension(self, poly, name, names, embedding)

    @cached_method
    def is_prime_field(self):
        """
        Return ``True`` if the order is prime.

        EXAMPLES::

            sage: Zmod(7).is_prime_field()
            True
            sage: Zmod(8).is_prime_field()
            False
        """
        return self.__order.is_prime()

    def _precompute_table(self):
        """
        Computes a table of elements so that elements are unique.

        EXAMPLES::

            sage: R = Zmod(500); R._precompute_table()
            sage: R(7) + R(13) is R(3) + R(17)
            True
        """
        self._pyx_order.precompute_table(self)

    def list_of_elements_of_multiplicative_group(self):
        """
        Return a list of all invertible elements, as python ints.

        EXAMPLES::

            sage: R = Zmod(12)
            sage: L = R.list_of_elements_of_multiplicative_group(); L
            [1, 5, 7, 11]
            sage: type(L[0])
            <type 'int'>
        """
        import sage.rings.fast_arith as a
        if self.__order <= 46340:   # todo: don't hard code
            gcd = a.arith_int().gcd_int
        elif self.__order <= 2147483647:   # todo: don't hard code
            gcd = a.arith_llong().gcd_longlong
        else:
            raise MemoryError("creating the list would exhaust memory")
        N = self.__order
        H = [i for i in range(N) if gcd(i, N) == 1]
        return H

    @cached_method
    def multiplicative_subgroups(self):
        r"""
        Return generators for each subgroup of `(\ZZ/N\ZZ)^*`.

        EXAMPLES::

            sage: Integers(5).multiplicative_subgroups()
            ((2,), (4,), ())
            sage: Integers(15).multiplicative_subgroups()
            ((11, 7), (4, 11), (8,), (11,), (14,), (7,), (4,), ())
            sage: Integers(2).multiplicative_subgroups()
            ((),)
            sage: len(Integers(341).multiplicative_subgroups())
            80

        TESTS::

            sage: IntegerModRing(1).multiplicative_subgroups()
            ((),)
            sage: IntegerModRing(2).multiplicative_subgroups()
            ((),)
            sage: IntegerModRing(3).multiplicative_subgroups()
            ((2,), ())
        """
        return tuple(tuple(g.value() for g in H.gens())
                     for H in self.unit_group().subgroups())

    def is_finite(self):
        r"""
        Return ``True`` since `\ZZ/N\ZZ` is finite for all positive `N`.

        EXAMPLES::

            sage: R = IntegerModRing(18)
            sage: R.is_finite()
            True
        """
        return True

    def is_integral_domain(self, proof=None):
        """
        Return ``True`` if and only if the order of ``self`` is prime.

        EXAMPLES::

            sage: Integers(389).is_integral_domain()
            True
            sage: Integers(389^2).is_integral_domain()
            False

        TESTS:

        Check that :trac:`17453` is fixed::

            sage: R = Zmod(5)
            sage: R in IntegralDomains()
            True
        """
        return self.is_field(proof)

    def is_unique_factorization_domain(self, proof=None):
        """
        Return ``True`` if and only if the order of ``self`` is prime.

        EXAMPLES::

            sage: Integers(389).is_unique_factorization_domain()
            True
            sage: Integers(389^2).is_unique_factorization_domain()
            False
        """
        return self.is_field(proof)

    @cached_method
    def is_field(self, proof=None):
        r"""
        Return True precisely if the order is prime.

        INPUT:

        - ``proof`` (optional bool or None, default None):
          If ``False``, then test whether the category of the quotient
          is a subcategory of ``Fields()``, or do a probabilistic
          primality test. If ``None``, then test the category and then
          do a primality test according to the global arithmetic proof
          settings. If True, do a deterministic primality test.

        If it is found (perhaps probabilistically) that the ring is a field,
        then the category of the ring is refined to include the category
        of fields. This may change the Python class of the ring!

        EXAMPLES::

            sage: R = IntegerModRing(18)
            sage: R.is_field()
            False
            sage: FF = IntegerModRing(17)
            sage: FF.is_field()
            True

        By :trac:`15229`, the category of the ring is refined,
        if it is found that the ring is in fact a field::

            sage: R = IntegerModRing(127)
            sage: R.category()
            Join of Category of finite commutative rings
                and Category of subquotients of monoids
                and Category of quotients of semigroups
                and Category of finite enumerated sets
            sage: R.is_field()
            True
            sage: R.category()
            Join of Category of finite fields
                and Category of subquotients of monoids
                and Category of quotients of semigroups

        It is possible to mistakenly put `\ZZ/n\ZZ` into the category of fields.
        In this case, :meth:`is_field` will return True without performing a
        primality check. However, if the optional argument `proof=True` is
        provided, primality is tested and the mistake is uncovered in a warning
        message::

            sage: R = IntegerModRing(21, is_field=True)
            sage: R.is_field()
            True
            sage: R.is_field(proof=True)
            Traceback (most recent call last):
            ...
            ValueError: THIS SAGE SESSION MIGHT BE SERIOUSLY COMPROMISED!
            The order 21 is not prime, but this ring has been put
            into the category of fields. This may already have consequences
            in other parts of Sage. Either it was a mistake of the user,
            or a probabilistic primality test has failed.
            In the latter case, please inform the developers.

        """
        from sage.categories.fields import Fields
        if not proof:
            if self.category().is_subcategory(Fields()):
                return True
        is_prime = self.order().is_prime(proof=proof)
        if is_prime:
            self._refine_category_(Fields())
            self._factory_data[3]['category'] = Fields()
        else:
            if self.category().is_subcategory(Fields()):
                raise ValueError("""THIS SAGE SESSION MIGHT BE SERIOUSLY COMPROMISED!
The order {} is not prime, but this ring has been put
into the category of fields. This may already have consequences
in other parts of Sage. Either it was a mistake of the user,
or a probabilistic primality test has failed.
In the latter case, please inform the developers.""".format(self.order()))
        return is_prime

    @cached_method
    def field(self):
        """
        If this ring is a field, return the corresponding field as a finite
        field, which may have extra functionality and structure. Otherwise,
        raise a ``ValueError``.

        EXAMPLES::

            sage: R = Integers(7); R
            Ring of integers modulo 7
            sage: R.field()
            Finite Field of size 7
            sage: R = Integers(9)
            sage: R.field()
            Traceback (most recent call last):
            ...
            ValueError: self must be a field
        """
        try:
            return self.__field
        except AttributeError:
            if not self.is_field():
                raise ValueError("self must be a field")
            from . import finite_field_constructor
            k = finite_field_constructor.FiniteField(self.order())
            self.__field = k
            return k

    def _pseudo_fraction_field(self):
        """
        If ``self`` is composite, we may still want to do division by elements
        of ``self``.

        EXAMPLES::

            sage: Integers(15).fraction_field()
            Traceback (most recent call last):
            ...
            TypeError: self must be an integral domain.
            sage: Integers(15)._pseudo_fraction_field()
            Ring of integers modulo 15
            sage: R.<x> = Integers(15)[]
            sage: (x+5)/2
            8*x + 10

        This should be very fast::

            sage: R.<x> = Integers(next_prime(10^101)*next_prime(10^100))[]
            sage: x / R.base_ring()(2)
            500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013365000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000401*x
        """
        return self

    @cached_method
    def multiplicative_group_is_cyclic(self):
        """
        Return ``True`` if the multiplicative group of this field is cyclic.
        This is the case exactly when the order is less than 8, a power
        of an odd prime, or twice a power of an odd prime.

        EXAMPLES::

            sage: R = Integers(7); R
            Ring of integers modulo 7
            sage: R.multiplicative_group_is_cyclic()
            True
            sage: R = Integers(9)
            sage: R.multiplicative_group_is_cyclic()
            True
            sage: Integers(8).multiplicative_group_is_cyclic()
            False
            sage: Integers(4).multiplicative_group_is_cyclic()
            True
            sage: Integers(25*3).multiplicative_group_is_cyclic()
            False

        We test that :trac:`5250` is fixed::

            sage: Integers(162).multiplicative_group_is_cyclic()
            True
        """
        n = self.order()
        if n < 8:
            return True

        if n % 4 == 0:
            return False # know n > 7, so n=4 case not a problem
        if n % 4 == 2:
            n = n // 2

        return n.is_prime_power()

    @cached_method
    def multiplicative_generator(self):
        """
        Return a generator for the multiplicative group of this ring,
        assuming the multiplicative group is cyclic.

        Use the unit_gens function to obtain generators even in the
        non-cyclic case.

        EXAMPLES::

            sage: R = Integers(7); R
            Ring of integers modulo 7
            sage: R.multiplicative_generator()
            3
            sage: R = Integers(9)
            sage: R.multiplicative_generator()
            2
            sage: Integers(8).multiplicative_generator()
            Traceback (most recent call last):
            ...
            ValueError: multiplicative group of this ring is not cyclic
            sage: Integers(4).multiplicative_generator()
            3
            sage: Integers(25*3).multiplicative_generator()
            Traceback (most recent call last):
            ...
            ValueError: multiplicative group of this ring is not cyclic
            sage: Integers(25*3).unit_gens()
            (26, 52)
            sage: Integers(162).unit_gens()
            (83,)
        """
        try:
            return self.__mult_gen
        except AttributeError:
            if self.is_field():
                a = self(self.field().multiplicative_generator())
                self.__mult_gen = a
                return a
            if self.multiplicative_group_is_cyclic():
                v = self.unit_gens()
                if len(v) != 1:
                    raise ArithmeticError
                return v[0]

            raise ValueError("multiplicative group of this ring is not cyclic")

    def quadratic_nonresidue(self):
        """
        Return a quadratic non-residue in ``self``.

        EXAMPLES::

            sage: R = Integers(17)
            sage: R.quadratic_nonresidue()
            3
            sage: R(3).is_square()
            False
        """
        try:
            return self._nonresidue
        except AttributeError:
            for a in self:
                if not a.is_square():
                    self._nonresidue = a
                    return a

    def square_roots_of_one(self):
        """
        Return all square roots of 1 in self, i.e., all solutions to
        `x^2 - 1 = 0`.

        OUTPUT:

        The square roots of 1 in ``self`` as a tuple.

        EXAMPLES::

            sage: R = Integers(2^10)
            sage: [x for x in R if x^2 == 1]
            [1, 511, 513, 1023]
            sage: R.square_roots_of_one()
            (1, 511, 513, 1023)

        ::

            sage: v = Integers(9*5).square_roots_of_one(); v
            (1, 19, 26, 44)
            sage: [x^2 for x in v]
            [1, 1, 1, 1]
            sage: v = Integers(9*5*8).square_roots_of_one(); v
            (1, 19, 71, 89, 91, 109, 161, 179, 181, 199, 251, 269, 271, 289, 341, 359)
            sage: [x^2 for x in v]
            [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
        """
        try:
            return self.__square_roots_of_one
        except AttributeError:
            pass
        n = self.__order
        if n.is_prime_power():
            if n % 2 == 0:
                # power of 2
                if n == 2:
                    v = [self(1)]
                elif n == 4:
                    v = [self(1), self(3)]
                else: # n >= 8
                    half_ord = n//2
                    v = [self(1), self(-1), self(half_ord-1), self(half_ord+1)]
            else:
                v = [self(1), self(-1)]
        else:
            # Reduce to the prime power case.
            F = self.factored_order()
            vmod = []
            moduli = []
            for p, e in F:
                k = p**e
                R = IntegerModRing(p**e)
                w = [self(x) for x in R.square_roots_of_one()]
                vmod.append(w)
                moduli.append(k)
            # Now combine in all possible ways using the CRT
            basis = CRT_basis(moduli)
            from sage.misc.mrange import cartesian_product_iterator
            v = []
            for x in cartesian_product_iterator(vmod):
                # x is a specific choice of roots modulo each prime power divisor
                a = sum([basis[i]*x[i] for i in range(len(x))])
                v.append(a)
            #end for
        #end if

        v.sort()
        v = tuple(v)
        self.__square_roots_of_one = v
        return v

    @cached_method
    def factored_order(self):
        """
        EXAMPLES::

            sage: R = IntegerModRing(18)
            sage: FF = IntegerModRing(17)
            sage: R.factored_order()
            2 * 3^2
            sage: FF.factored_order()
            17
        """
        return factor(self.__order, int_=(self.__order < 2**31))

    def factored_unit_order(self):
        """
        Return a list of :class:`Factorization` objects, each the factorization
        of the order of the units in a `\ZZ / p^n \ZZ` component of this group
        (using the Chinese Remainder Theorem).

        EXAMPLES::

            sage: R = Integers(8*9*25*17*29)
            sage: R.factored_unit_order()
            [2^2, 2 * 3, 2^2 * 5, 2^4, 2^2 * 7]
        """
        ans = []
        from sage.structure.factorization import Factorization
        for p, e in self.factored_order():
            ans.append(Factorization([(p,e-1)]) * factor(p-1, int_=(self.__order < 2**31)))
        return ans

    def characteristic(self):
        """
        EXAMPLES::

            sage: R = IntegerModRing(18)
            sage: FF = IntegerModRing(17)
            sage: FF.characteristic()
            17
            sage: R.characteristic()
            18
        """
        return self.__order

    def _repr_(self):
        """
        String representation.

        EXAMPLES::

            sage: Zmod(87)
            Ring of integers modulo 87
        """
        return "Ring of integers modulo {}".format(self.__order)

    def _latex_(self):
        r"""
        Latex representation.

        EXAMPLES::

            sage: latex(Zmod(87))
            \ZZ/87\ZZ
        """
        return "\\ZZ/{}\\ZZ".format(self.__order)

    def modulus(self):
        r"""
        Return the polynomial `x - 1` over this ring.

        .. NOTE::

           This function exists for consistency with the finite-field
           modulus function.

        EXAMPLES::

            sage: R = IntegerModRing(18)
            sage: R.modulus()
            x + 17
            sage: R = IntegerModRing(17)
            sage: R.modulus()
            x + 16
        """
        try:
            return self.__modulus
        except AttributeError:
            x = self['x'].gen()
            self.__modulus = x - 1
            return self.__modulus

    def order(self):
        """
        Return the order of this ring.

        EXAMPLES::

            sage: Zmod(87).order()
            87
        """
        return self.__order

    def cardinality(self):
        """
        Return the cardinality of this ring.

        EXAMPLES::

            sage: Zmod(87).cardinality()
            87
        """
        return self.order()

    def _pari_order(self):
        """
        Return the pari integer representing the order of this ring.

        EXAMPLES::

            sage: Zmod(87)._pari_order()
            87
        """
        try:
            return self.__pari_order
        except AttributeError:
            self.__pari_order = pari(self.order())
            return self.__pari_order

    def _element_constructor_(self, x):
        """
        TESTS::

            sage: K2 = GF(2)
            sage: K3 = GF(3)
            sage: K8 = GF(8,'a')
            sage: K8(5) # indirect doctest
            1
            sage: K8('a+1')
            a + 1
            sage: K8(K2(1))
            1

        The following test refers to :trac:`6468`::

            sage: class foo_parent(Parent):
            ...       pass
            sage: class foo(RingElement):
            ...       def lift(self):
            ...           raise PariError
            sage: P = foo_parent()
            sage: F = foo(P)
            sage: GF(2)(F)
            Traceback (most recent call last):
            ...
            TypeError: error coercing to finite field

        The following test refers to :trac:`8970`::

            sage: R = Zmod(13); a = R(2)
            sage: a == R(gap(a))
            True

        """
        try:
            return integer_mod.IntegerMod(self, x)
        except (NotImplementedError, PariError):
            raise TypeError("error coercing to finite field")
        except TypeError:
            if sage.interfaces.gap.is_GapElement(x):
                from sage.interfaces.gap import intmod_gap_to_sage
                y = intmod_gap_to_sage(x)
                return integer_mod.IntegerMod(self, y)
            raise # Continue up with the original TypeError

    def __iter__(self):
        """
        EXAMPLES::

            sage: R = IntegerModRing(3)
            sage: for i in R:
            ....:     print(i)
            0
            1
            2
            sage: L = [i for i in R]
            sage: L[0].parent()
            Ring of integers modulo 3
        """
        i = 0
        order = int(self.__order)
        while i < order:
            yield self(i)
            i = i + 1

    def _coerce_map_from_(self, S):
        """
        EXAMPLES::

            sage: R = Integers(15)
            sage: f = R.coerce_map_from(Integers(450)); f # indirect doctest
            Natural morphism:
              From: Ring of integers modulo 450
              To:   Ring of integers modulo 15
            sage: f(-1)
            14
            sage: f = R.coerce_map_from(int); f
            Native morphism:
              From: Set of Python objects of type 'int'
              To:   Ring of integers modulo 15
            sage: f(-1r)
            14
            sage: f = R.coerce_map_from(ZZ); f
            Natural morphism:
              From: Integer Ring
              To:   Ring of integers modulo 15
            sage: f(-1)
            14
            sage: f = R.coerce_map_from(Integers(10)); print(f)
            None
            sage: f = R.coerce_map_from(QQ); print(f)
            None

            sage: R = IntegerModRing(17)
            sage: a = R(3)
            sage: b = R._coerce_(3)
            sage: b
            3
            sage: a==b
            True

        This is allowed::

            sage: R(2/3)
            12

        But this is not, since there is no (canonical or not!) ring
        homomorphism from `\QQ` to `\GF{17}`.

        ::

            sage: R._coerce_(2/3)
            Traceback (most recent call last):
            ...
            TypeError: no canonical coercion from Rational Field to Ring of integers modulo 17

        We do not allow the coercion ``GF(p) -> Z/pZ``, because in case of a
        canonical isomorphism, there is a coercion map in only one
        direction, i.e., to the object in the smaller category.
        """
        if S is int:
            return integer_mod.Int_to_IntegerMod(self)
        elif S is integer_ring.ZZ:
            return integer_mod.Integer_to_IntegerMod(self)
        elif isinstance(S, IntegerModRing_generic):
            if isinstance(S, ring.Field):
                return None
            try:
                return integer_mod.IntegerMod_to_IntegerMod(S, self)
            except TypeError:
                pass
        to_ZZ = integer_ring.ZZ._internal_coerce_map_from(S)
        if to_ZZ is not None:
            return integer_mod.Integer_to_IntegerMod(self) * to_ZZ

    def __cmp__(self, other):
        """
        EXAMPLES::

            sage: Z11 = IntegerModRing(11); Z11
            Ring of integers modulo 11
            sage: Z12 = IntegerModRing(12); Z12
            Ring of integers modulo 12
            sage: Z13 = IntegerModRing(13); Z13
            Ring of integers modulo 13
            sage: F = GF(11); F
            Finite Field of size 11
            sage: Z11 == Z11, Z11 == Z12, Z11 == Z13, Z11 == F
            (True, False, False, False)

        In :trac:`15229`, the following was implemented::

            sage: R1 = IntegerModRing(5)
            sage: R2 = IntegerModRing(5, is_field=True)
            sage: R1 is R2    # used to return False
            True
            sage: R2 == GF(5)
            False

        """
        # We want that GF(p) and IntegerModRing(p) evaluate unequal.
        # However, we can not just compare the types, since the
        # choice of a different category also changes the type.
        # But if we go to the base class, we avoid the influence
        # of the category.
        try:
            c = cmp(other.__class__.__base__, self.__class__.__base__)
        except AttributeError: # __base__ does not always exists
            c = cmp(type(other), type(self))
        if c:
            return c
        return cmp(self.__order, other.__order)

    def unit_gens(self, **kwds):
        r"""
        Returns generators for the unit group `(\ZZ/N\ZZ)^*`.

        We compute the list of generators using a deterministic algorithm, so
        the generators list will always be the same. For each odd prime divisor
        of `N` there will be exactly one corresponding generator; if `N` is
        even there will be 0, 1 or 2 generators according to whether 2 divides
        `N` to order 1, 2 or `\geq 3`.

        OUTPUT:

        A tuple containing the units of ``self``.

        EXAMPLES::

            sage: R = IntegerModRing(18)
            sage: R.unit_gens()
            (11,)
            sage: R = IntegerModRing(17)
            sage: R.unit_gens()
            (3,)
            sage: IntegerModRing(next_prime(10^30)).unit_gens()
            (5,)

        The choice of generators is affected by the optional keyword
        ``algorithm``; this can be ``'sage'`` (default) or ``'pari'``.
        See :meth:`unit_group` for details.

            sage: A = Zmod(55)
            sage: A.unit_gens(algorithm='sage')
            (12, 46)
            sage: A.unit_gens(algorithm='pari')
            (2, 21)

        TESTS::

            sage: IntegerModRing(2).unit_gens()
            ()
            sage: IntegerModRing(4).unit_gens()
            (3,)
            sage: IntegerModRing(8).unit_gens()
            (7, 5)

        """
        return self.unit_group(**kwds).gens_values()

    def unit_group_exponent(self):
        """
        EXAMPLES::

            sage: R = IntegerModRing(17)
            sage: R.unit_group_exponent()
            16
            sage: R = IntegerModRing(18)
            sage: R.unit_group_exponent()
            6
        """
        return self.unit_group().exponent()

    def unit_group_order(self):
        """
        Return the order of the unit group of this residue class ring.

        EXAMPLES::

            sage: R = Integers(500)
            sage: R.unit_group_order()
            200
        """
        return self.unit_group().order()

    @cached_method
    def unit_group(self, algorithm='sage'):
        r"""
        Return the unit group of ``self``.

        INPUT:

        - ``self`` -- the ring `\ZZ/n\ZZ` for a positive integer `n`

        - ``algorithm`` -- either ``'sage'`` (default) or ``'pari'``

        OUTPUT:

        The unit group of ``self``.  This is a finite Abelian group
        equipped with a distinguished set of generators, which is
        computed using a deterministic algorithm depending on the
        ``algorithm`` parameter.

        - If ``algorithm == 'sage'``, the generators correspond to the
          prime factors `p \mid n` (one generator for each odd `p`;
          the number of generators for `p = 2` is 0, 1 or 2 depending
          on the order to which 2 divides `n`).

        - If ``algorithm == 'pari'``, the generators are chosen such
          that their orders form a decreasing sequence with respect to
          divisibility.

        EXAMPLES:

        The output of the algorithms ``'sage'`` and ``'pari'`` can
        differ in various ways.  In the following example, the same
        cyclic factors are computed, but in a different order::

            sage: A = Zmod(15)
            sage: G = A.unit_group(); G
            Multiplicative Abelian group isomorphic to C2 x C4
            sage: G.gens_values()
            (11, 7)
            sage: H = A.unit_group(algorithm='pari'); H
            Multiplicative Abelian group isomorphic to C4 x C2
            sage: H.gens_values()
            (7, 11)

        Here are two examples where the cyclic factors are isomorphic,
        but are ordered differently and have different generators::

            sage: A = Zmod(40)
            sage: G = A.unit_group(); G
            Multiplicative Abelian group isomorphic to C2 x C2 x C4
            sage: G.gens_values()
            (31, 21, 17)
            sage: H = A.unit_group(algorithm='pari'); H
            Multiplicative Abelian group isomorphic to C4 x C2 x C2
            sage: H.gens_values()
            (17, 31, 21)

            sage: A = Zmod(192)
            sage: G = A.unit_group(); G
            Multiplicative Abelian group isomorphic to C2 x C16 x C2
            sage: G.gens_values()
            (127, 133, 65)
            sage: H = A.unit_group(algorithm='pari'); H
            Multiplicative Abelian group isomorphic to C16 x C2 x C2
            sage: H.gens_values()
            (133, 127, 65)

        In the following examples, the cyclic factors are not even
        isomorphic::

            sage: A = Zmod(319)
            sage: A.unit_group()
            Multiplicative Abelian group isomorphic to C10 x C28
            sage: A.unit_group(algorithm='pari')
            Multiplicative Abelian group isomorphic to C140 x C2

            sage: A = Zmod(30.factorial())
            sage: A.unit_group()
            Multiplicative Abelian group isomorphic to C2 x C16777216 x C3188646 x C62500 x C2058 x C110 x C156 x C16 x C18 x C22 x C28
            sage: A.unit_group(algorithm='pari')
            Multiplicative Abelian group isomorphic to C20499647385305088000000 x C55440 x C12 x C12 x C4 x C2 x C2 x C2 x C2 x C2 x C2

        TESTS:

        We test the cases where the unit group is trivial::

            sage: A = Zmod(1)
            sage: A.unit_group()
            Trivial Abelian group
            sage: A.unit_group(algorithm='pari')
            Trivial Abelian group
            sage: A = Zmod(2)
            sage: A.unit_group()
            Trivial Abelian group
            sage: A.unit_group(algorithm='pari')
            Trivial Abelian group

            sage: Zmod(3).unit_group(algorithm='bogus')
            Traceback (most recent call last):
            ...
            ValueError: unknown algorithm 'bogus' for computing the unit group

        """
        from sage.groups.abelian_gps.values import AbelianGroupWithValues
        if algorithm == 'sage':
            n = self.order()
            gens = []
            orders = []
            for p, r in self.factored_order():
                m = n/(p**r)
                for g, o in _unit_gens_primepowercase(p, r):
                    x = g.crt(integer_mod.Mod(1, m))
                    gens.append(x)
                    orders.append(o)
        elif algorithm == 'pari':
            _, orders, gens = self.order()._pari_().znstar()
            gens = map(self, gens)
            orders = map(integer.Integer, orders)
        else:
            raise ValueError('unknown algorithm %r for computing the unit group' % algorithm)
        return AbelianGroupWithValues(gens, orders, values_group=self)

    def random_element(self, bound=None):
        """
        Return a random element of this ring.

        INPUT:

        - ``bound``, a positive integer or ``None`` (the default). Is given,
          return  the coercion of an integer in the interval
          ``[-bound, bound]`` into this ring.

        EXAMPLES::

            sage: R = IntegerModRing(18)
            sage: R.random_element()
            2

        We test ``bound``-option::

            sage: R.random_element(2) in [R(16), R(17), R(0), R(1), R(2)]
            True
        """
        if bound is not None:
            return ring.CommutativeRing.random_element(self, bound)
        a = random.randint(0,self.order()-1)
        return self(a)

    #######################################################
    # Suppose for interfaces
    #######################################################
    def _gap_init_(self):
        """
        EXAMPLES::

            sage: R = Integers(12345678900)
            sage: R
            Ring of integers modulo 12345678900
            sage: gap(R) # indirect doctest
            (Integers mod 12345678900)
        """
        return 'ZmodnZ({})'.format(self.order())

    def _magma_init_(self, magma):
        """
        EXAMPLES::

            sage: R = Integers(12345678900)
            sage: R
            Ring of integers modulo 12345678900
            sage: magma(R) # indirect doctest, optional - magma
            Residue class ring of integers modulo 12345678900
        """
        return 'Integers({})'.format(self.order())

    def degree(self):
        """
        Return 1.

        EXAMPLE::

            sage: R = Integers(12345678900)
            sage: R.degree()
            1
        """
        return integer.Integer(1)

Zmod = IntegerModRing
Integers = IntegerModRing

# Register unpickling methods for backward compatibility.

from sage.structure.sage_object import register_unpickle_override
register_unpickle_override('sage.rings.integer_mod_ring', 'IntegerModRing_generic', IntegerModRing_generic)

def crt(v):
    """
    INPUT:

    - ``v`` -- (list) a lift of elements of ``rings.IntegerMod(n)``, for
      various coprime moduli ``n``

    EXAMPLES::

        sage: from sage.rings.finite_rings.integer_mod_ring import crt
        sage: crt([mod(3, 8),mod(1,19),mod(7, 15)])
        1027
    """
    if len(v) == 0:
        return IntegerModRing(1)(1)
    x = v[0]
    for i in range(1,len(v)):
        x = x.crt(v[i])
    return x

