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"""Implementation of :class:`FiniteField` class. """
import operator
from sympy.external.gmpy import GROUND_TYPES
from sympy.utilities.decorator import doctest_depends_on
from sympy.core.numbers import int_valued
from sympy.polys.domains.field import Field
from sympy.polys.domains.modularinteger import ModularIntegerFactory
from sympy.polys.domains.simpledomain import SimpleDomain
from sympy.polys.galoistools import gf_zassenhaus, gf_irred_p_rabin
from sympy.polys.polyerrors import CoercionFailed
from sympy.utilities import public
from sympy.polys.domains.groundtypes import SymPyInteger
if GROUND_TYPES == 'flint':
__doctest_skip__ = ['FiniteField']
if GROUND_TYPES == 'flint':
import flint
# Don't use python-flint < 0.5.0 because nmod was missing some features in
# previous versions of python-flint and fmpz_mod was not yet added.
_major, _minor, *_ = flint.__version__.split('.')
if (int(_major), int(_minor)) < (0, 5):
flint = None
else:
flint = None
def _modular_int_factory_nmod(mod):
# nmod only recognises int
index = operator.index
mod = index(mod)
nmod = flint.nmod
nmod_poly = flint.nmod_poly
# flint's nmod is only for moduli up to 2^64-1 (on a 64-bit machine)
try:
nmod(0, mod)
except OverflowError:
return None, None
def ctx(x):
try:
return nmod(x, mod)
except TypeError:
return nmod(index(x), mod)
def poly_ctx(cs):
return nmod_poly(cs, mod)
return ctx, poly_ctx
def _modular_int_factory_fmpz_mod(mod):
index = operator.index
fctx = flint.fmpz_mod_ctx(mod)
fctx_poly = flint.fmpz_mod_poly_ctx(mod)
fmpz_mod_poly = flint.fmpz_mod_poly
def ctx(x):
try:
return fctx(x)
except TypeError:
# x might be Integer
return fctx(index(x))
def poly_ctx(cs):
return fmpz_mod_poly(cs, fctx_poly)
return ctx, poly_ctx
def _modular_int_factory(mod, dom, symmetric, self):
# Convert the modulus to ZZ
try:
mod = dom.convert(mod)
except CoercionFailed:
raise ValueError('modulus must be an integer, got %s' % mod)
ctx, poly_ctx, is_flint = None, None, False
# Don't use flint if the modulus is not prime as it often crashes.
if flint is not None and mod.is_prime():
is_flint = True
# Try to use flint's nmod first
ctx, poly_ctx = _modular_int_factory_nmod(mod)
if ctx is None:
# Use fmpz_mod for larger moduli
ctx, poly_ctx = _modular_int_factory_fmpz_mod(mod)
if ctx is None:
# Use the Python implementation if flint is not available or the
# modulus is not prime.
ctx = ModularIntegerFactory(mod, dom, symmetric, self)
poly_ctx = None # not used
return ctx, poly_ctx, is_flint
@public
@doctest_depends_on(modules=['python', 'gmpy'])
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)
ctx, poly_ctx, is_flint = _modular_int_factory(mod, dom, symmetric, self)
self.dtype = ctx
self._poly_ctx = poly_ctx
self._is_flint = is_flint
self.zero = self.dtype(0)
self.one = self.dtype(1)
self.dom = dom
self.mod = mod
self.sym = symmetric
self._tp = type(self.zero)
@property
def tp(self):
return self._tp
@property
def is_Field(self):
is_field = getattr(self, '_is_field', None)
if is_field is None:
from sympy.ntheory.primetest import isprime
self._is_field = is_field = isprime(self.mod)
return is_field
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(self.to_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 int_valued(a):
return self.dtype(self.dom.dtype(int(a)))
else:
raise CoercionFailed("expected an integer, got %s" % a)
def to_int(self, a):
"""Convert ``val`` to a Python ``int`` object. """
aval = int(a)
if self.sym and aval > self.mod // 2:
aval -= self.mod
return aval
def is_positive(self, a):
"""Returns True if ``a`` is positive. """
return bool(a)
def is_nonnegative(self, a):
"""Returns True if ``a`` is non-negative. """
return True
def is_negative(self, a):
"""Returns True if ``a`` is negative. """
return False
def is_nonpositive(self, a):
"""Returns True if ``a`` is non-positive. """
return not a
def from_FF(K1, a, K0=None):
"""Convert ``ModularInteger(int)`` to ``dtype``. """
return K1.dtype(K1.dom.from_ZZ(int(a), K0.dom))
def from_FF_python(K1, a, K0=None):
"""Convert ``ModularInteger(int)`` to ``dtype``. """
return K1.dtype(K1.dom.from_ZZ_python(int(a), 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))
def is_square(self, a):
"""Returns True if ``a`` is a quadratic residue modulo p. """
# a is not a square <=> x**2-a is irreducible
poly = [int(x) for x in [self.one, self.zero, -a]]
return not gf_irred_p_rabin(poly, self.mod, self.dom)
def exsqrt(self, a):
"""Square root modulo p of ``a`` if it is a quadratic residue.
Explanation
===========
Always returns the square root that is no larger than ``p // 2``.
"""
# x**2-a is not square-free if a=0 or the field is characteristic 2
if self.mod == 2 or a == 0:
return a
# Otherwise, use square-free factorization routine to factorize x**2-a
poly = [int(x) for x in [self.one, self.zero, -a]]
for factor in gf_zassenhaus(poly, self.mod, self.dom):
if len(factor) == 2 and factor[1] <= self.mod // 2:
return self.dtype(factor[1])
return None
FF = GF = FiniteField
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