ai-content-maker/.venv/Lib/site-packages/sympy/polys/domains/finitefield.py

"""Implementation of :class:`FiniteField` class. """


from sympy.polys.domains.field import Field

from sympy.polys.domains.modularinteger import ModularIntegerFactory
from sympy.polys.domains.simpledomain import SimpleDomain
from sympy.polys.polyerrors import CoercionFailed
from sympy.utilities import public
from sympy.polys.domains.groundtypes import SymPyInteger

@public
class FiniteField(Field, SimpleDomain):
    r"""Finite field of prime order :ref:`GF(p)`

    A :ref:`GF(p)` domain represents a `finite field`_ `\mathbb{F}_p` of prime
    order as :py:class:`~.Domain` in the domain system (see
    :ref:`polys-domainsintro`).

    A :py:class:`~.Poly` created from an expression with integer
    coefficients will have the domain :ref:`ZZ`. However, if the ``modulus=p``
    option is given then the domain will be a finite field instead.

    >>> from sympy import Poly, Symbol
    >>> x = Symbol('x')
    >>> p = Poly(x**2 + 1)
    >>> p
    Poly(x**2 + 1, x, domain='ZZ')
    >>> p.domain
    ZZ
    >>> p2 = Poly(x**2 + 1, modulus=2)
    >>> p2
    Poly(x**2 + 1, x, modulus=2)
    >>> p2.domain
    GF(2)

    It is possible to factorise a polynomial over :ref:`GF(p)` using the
    modulus argument to :py:func:`~.factor` or by specifying the domain
    explicitly. The domain can also be given as a string.

    >>> from sympy import factor, GF
    >>> factor(x**2 + 1)
    x**2 + 1
    >>> factor(x**2 + 1, modulus=2)
    (x + 1)**2
    >>> factor(x**2 + 1, domain=GF(2))
    (x + 1)**2
    >>> factor(x**2 + 1, domain='GF(2)')
    (x + 1)**2

    It is also possible to use :ref:`GF(p)` with the :py:func:`~.cancel`
    and :py:func:`~.gcd` functions.

    >>> from sympy import cancel, gcd
    >>> cancel((x**2 + 1)/(x + 1))
    (x**2 + 1)/(x + 1)
    >>> cancel((x**2 + 1)/(x + 1), domain=GF(2))
    x + 1
    >>> gcd(x**2 + 1, x + 1)
    1
    >>> gcd(x**2 + 1, x + 1, domain=GF(2))
    x + 1

    When using the domain directly :ref:`GF(p)` can be used as a constructor
    to create instances which then support the operations ``+,-,*,**,/``

    >>> from sympy import GF
    >>> K = GF(5)
    >>> K
    GF(5)
    >>> x = K(3)
    >>> y = K(2)
    >>> x
    3 mod 5
    >>> y
    2 mod 5
    >>> x * y
    1 mod 5
    >>> x / y
    4 mod 5

    Notes
    =====

    It is also possible to create a :ref:`GF(p)` domain of **non-prime**
    order but the resulting ring is **not** a field: it is just the ring of
    the integers modulo ``n``.

    >>> K = GF(9)
    >>> z = K(3)
    >>> z
    3 mod 9
    >>> z**2
    0 mod 9

    It would be good to have a proper implementation of prime power fields
    (``GF(p**n)``) but these are not yet implemented in SymPY.

    .. _finite field: https://en.wikipedia.org/wiki/Finite_field
    """

    rep = 'FF'
    alias = 'FF'

    is_FiniteField = is_FF = True
    is_Numerical = True

    has_assoc_Ring = False
    has_assoc_Field = True

    dom = None
    mod = None

    def __init__(self, mod, symmetric=True):
        from sympy.polys.domains import ZZ
        dom = ZZ

        if mod <= 0:
            raise ValueError('modulus must be a positive integer, got %s' % mod)

        self.dtype = ModularIntegerFactory(mod, dom, symmetric, self)
        self.zero = self.dtype(0)
        self.one = self.dtype(1)
        self.dom = dom
        self.mod = mod

    def __str__(self):
        return 'GF(%s)' % self.mod

    def __hash__(self):
        return hash((self.__class__.__name__, self.dtype, self.mod, self.dom))

    def __eq__(self, other):
        """Returns ``True`` if two domains are equivalent. """
        return isinstance(other, FiniteField) and \
            self.mod == other.mod and self.dom == other.dom

    def characteristic(self):
        """Return the characteristic of this domain. """
        return self.mod

    def get_field(self):
        """Returns a field associated with ``self``. """
        return self

    def to_sympy(self, a):
        """Convert ``a`` to a SymPy object. """
        return SymPyInteger(int(a))

    def from_sympy(self, a):
        """Convert SymPy's Integer to SymPy's ``Integer``. """
        if a.is_Integer:
            return self.dtype(self.dom.dtype(int(a)))
        elif a.is_Float and int(a) == a:
            return self.dtype(self.dom.dtype(int(a)))
        else:
            raise CoercionFailed("expected an integer, got %s" % a)

    def from_FF(K1, a, K0=None):
        """Convert ``ModularInteger(int)`` to ``dtype``. """
        return K1.dtype(K1.dom.from_ZZ(a.val, K0.dom))

    def from_FF_python(K1, a, K0=None):
        """Convert ``ModularInteger(int)`` to ``dtype``. """
        return K1.dtype(K1.dom.from_ZZ_python(a.val, K0.dom))

    def from_ZZ(K1, a, K0=None):
        """Convert Python's ``int`` to ``dtype``. """
        return K1.dtype(K1.dom.from_ZZ_python(a, K0))

    def from_ZZ_python(K1, a, K0=None):
        """Convert Python's ``int`` to ``dtype``. """
        return K1.dtype(K1.dom.from_ZZ_python(a, K0))

    def from_QQ(K1, a, K0=None):
        """Convert Python's ``Fraction`` to ``dtype``. """
        if a.denominator == 1:
            return K1.from_ZZ_python(a.numerator)

    def from_QQ_python(K1, a, K0=None):
        """Convert Python's ``Fraction`` to ``dtype``. """
        if a.denominator == 1:
            return K1.from_ZZ_python(a.numerator)

    def from_FF_gmpy(K1, a, K0=None):
        """Convert ``ModularInteger(mpz)`` to ``dtype``. """
        return K1.dtype(K1.dom.from_ZZ_gmpy(a.val, K0.dom))

    def from_ZZ_gmpy(K1, a, K0=None):
        """Convert GMPY's ``mpz`` to ``dtype``. """
        return K1.dtype(K1.dom.from_ZZ_gmpy(a, K0))

    def from_QQ_gmpy(K1, a, K0=None):
        """Convert GMPY's ``mpq`` to ``dtype``. """
        if a.denominator == 1:
            return K1.from_ZZ_gmpy(a.numerator)

    def from_RealField(K1, a, K0):
        """Convert mpmath's ``mpf`` to ``dtype``. """
        p, q = K0.to_rational(a)

        if q == 1:
            return K1.dtype(K1.dom.dtype(p))


FF = GF = FiniteField