def __init__(self, q, name="a", modulus=None, repr="poly", cache=False):
"""
Initialize ``self``.
EXAMPLES::
sage: k.<a> = GF(2^3)
sage: j.<b> = GF(3^4)
sage: k == j
False
sage: GF(2^3,'a') == copy(GF(2^3,'a'))
True
sage: TestSuite(GF(2^3, 'a')).run()
"""
self._kwargs = {}
if repr not in ['int', 'log', 'poly']:
raise ValueError("Unknown representation %s" % repr)
q = Integer(q)
if q < 2:
raise ValueError("q must be a prime power")
F = q.factor()
if len(F) > 1:
raise ValueError("q must be a prime power")
p = F[0][0]
k = F[0][1]
if q >= 1 << 16:
raise ValueError("q must be < 2^16")
from constructor import GF
FiniteField.__init__(self, GF(p), name, normalize=False)
self._kwargs['repr'] = repr
self._kwargs['cache'] = cache
from sage.rings.polynomial.polynomial_element import is_Polynomial
if not is_Polynomial(modulus):
from sage.misc.superseded import deprecation
deprecation(
16930,
"constructing a FiniteField_givaro without giving a polynomial as modulus is deprecated, use the more general FiniteField constructor instead"
)
R = GF(p)['x']
if modulus is None or isinstance(modulus, str):
modulus = R.irreducible_element(k, algorithm=modulus)
else:
modulus = R(modulus)
self._cache = Cache_givaro(self, p, k, modulus, repr, cache)
self._modulus = modulus
def __init__(self, q, name="a", modulus=None, repr="poly", cache=False):
"""
Initialize ``self``.
EXAMPLES::
sage: k.<a> = GF(2^3)
sage: j.<b> = GF(3^4)
sage: k == j
False
sage: GF(2^3,'a') == copy(GF(2^3,'a'))
True
sage: TestSuite(GF(2^3, 'a')).run()
"""
self._kwargs = {}
if repr not in ['int', 'log', 'poly']:
raise ValueError("Unknown representation %s"%repr)
q = Integer(q)
if q < 2:
raise ValueError("q must be a prime power")
F = q.factor()
if len(F) > 1:
raise ValueError("q must be a prime power")
p = F[0][0]
k = F[0][1]
if q >= 1<<16:
raise ValueError("q must be < 2^16")
from constructor import GF
FiniteField.__init__(self, GF(p), name, normalize=False)
self._kwargs['repr'] = repr
self._kwargs['cache'] = cache
from sage.rings.polynomial.polynomial_element import is_Polynomial
if not is_Polynomial(modulus):
from sage.misc.superseded import deprecation
deprecation(16930, "constructing a FiniteField_givaro without giving a polynomial as modulus is deprecated, use the more general FiniteField constructor instead")
R = GF(p)['x']
if modulus is None or isinstance(modulus, str):
modulus = R.irreducible_element(k, algorithm=modulus)
else:
modulus = R(modulus)
self._cache = Cache_givaro(self, p, k, modulus, repr, cache)
self._modulus = modulus
def __init__(self, p, modulus, name=None):
"""
Create a finite field of characteristic `p` defined by the
polynomial ``modulus``, with distinguished generator called
``name``.
EXAMPLE::
sage: from sage.rings.finite_rings.finite_field_pari_ffelt import FiniteField_pari_ffelt
sage: R.<x> = PolynomialRing(GF(3))
sage: k = FiniteField_pari_ffelt(3, x^2 + 2*x + 2, 'a'); k
Finite Field in a of size 3^2
"""
n = modulus.degree()
if n < 2:
raise ValueError("the degree must be at least 2")
FiniteField.__init__(self, base=GF(p), names=name, normalize=True)
self._modulus = modulus
self._degree = n
self._kwargs = {}
self._gen_pari = modulus._pari_with_name(self._names[0]).ffgen()
self._zero_element = self.element_class(self, 0)
self._one_element = self.element_class(self, 1)
self._gen = self.element_class(self, self._gen_pari)
def polynomial(self, name=None):
"""
Return the irreducible characteristic polynomial of the
generator of this finite field, i.e., the polynomial `f(x)` so
elements of the finite field as elements modulo `f`.
EXAMPLES::
sage: k = FiniteField(17)
sage: k.polynomial('x')
x
sage: from sage.rings.finite_rings.finite_field_ext_pari import FiniteField_ext_pari
sage: k = FiniteField_ext_pari(9,'a')
sage: k.polynomial('x')
x^2 + 2*x + 2
"""
if name is None:
name = self.variable_name()
try:
return self.__polynomial[name]
except (AttributeError, KeyError):
from constructor import FiniteField as GF
R = GF(self.characteristic())[name]
f = R(self._pari_modulus())
try:
self.__polynomial[name] = f
except (KeyError, AttributeError):
self.__polynomial = {}
self.__polynomial[name] = f
return f
def prime_subfield(self):
r"""
Return the prime subfield `\GF{p}` of self if ``self`` is `\GF{p^n}`.
EXAMPLES::
sage: GF(3^4, 'b').prime_subfield()
Finite Field of size 3
sage: S.<b> = GF(5^2); S
Finite Field in b of size 5^2
sage: S.prime_subfield()
Finite Field of size 5
sage: type(S.prime_subfield())
<class 'sage.rings.finite_rings.finite_field_prime_modn.FiniteField_prime_modn_with_category'>
"""
try:
return self._prime_subfield
except AttributeError:
from constructor import GF
self._prime_subfield = GF(self.characteristic())
return self._prime_subfield
def __init__(self, q, name, modulus=None):
"""
Create finite field of order `q` with variable printed as name.
EXAMPLES::
sage: from sage.rings.finite_rings.finite_field_ext_pari import FiniteField_ext_pari
sage: k = FiniteField_ext_pari(9, 'a'); k
Finite Field in a of size 3^2
"""
if element_ext_pari.dynamic_FiniteField_ext_pariElement is None: element_ext_pari._late_import()
from constructor import FiniteField as GF
q = integer.Integer(q)
if q < 2:
raise ArithmeticError, "q must be a prime power"
from sage.structure.proof.all import arithmetic
proof = arithmetic()
if proof:
F = q.factor()
else:
from sage.rings.arith import is_pseudoprime_small_power
F = is_pseudoprime_small_power(q, get_data=True)
if len(F) != 1:
raise ArithmeticError, "q must be a prime power"
if F[0][1] > 1:
base_ring = GF(F[0][0])
else:
raise ValueError, "The size of the finite field must not be prime."
#base_ring = self
FiniteField_generic.__init__(self, base_ring, name, normalize=True)
self._kwargs = {}
self.__char = F[0][0]
self.__pari_one = pari.pari(1).Mod(self.__char)
self.__degree = integer.Integer(F[0][1])
self.__order = q
self.__is_field = True
if modulus is None or modulus == "default":
from conway_polynomials import exists_conway_polynomial
if exists_conway_polynomial(self.__char, self.__degree):
modulus = "conway"
else:
modulus = "random"
if isinstance(modulus,str):
if modulus == "conway":
from conway_polynomials import conway_polynomial
modulus = conway_polynomial(self.__char, self.__degree)
elif modulus == "random":
# The following is fast/deterministic, but has serious problems since
# it crashes on 64-bit machines, and I can't figure out why:
# self.__pari_modulus = pari.pari.finitefield_init(self.__char, self.__degree, self.variable_name())
# So instead we iterate through random polys until we find an irreducible one.
R = GF(self.__char)['x']
while True:
modulus = R.random_element(self.__degree)
modulus = modulus.monic()
if modulus.degree() == self.__degree and modulus.is_irreducible():
break
else:
raise ValueError("Modulus parameter not understood")
elif isinstance(modulus, (list, tuple)):
modulus = GF(self.__char)['x'](modulus)
elif sage.rings.polynomial.polynomial_element.is_Polynomial(modulus):
if modulus.parent() is not base_ring:
modulus = modulus.change_ring(base_ring)
else:
raise ValueError("Modulus parameter not understood")
self.__modulus = modulus
f = pari.pari(str(modulus))
self.__pari_modulus = f.subst(modulus.parent().variable_name(), 'a') * self.__pari_one
self.__gen = element_ext_pari.FiniteField_ext_pariElement(self, pari.pari('a'))
self._zero_element = self._element_constructor_(0)
self._one_element = self._element_constructor_(1)
def __init__(self, q, name, modulus=None):
"""
Create finite field of order `q` with variable printed as name.
EXAMPLES::
sage: k = FiniteField(9, 'a', impl='pari_mod'); k
Finite Field in a of size 3^2
"""
from sage.misc.superseded import deprecation
deprecation(17297, 'The "pari_mod" finite field implementation is deprecated')
if element_ext_pari.dynamic_FiniteField_ext_pariElement is None: element_ext_pari._late_import()
from constructor import FiniteField as GF
q = integer.Integer(q)
if q < 2:
raise ArithmeticError("q must be a prime power")
# note: the following call takes care of the fact that
# proof.arithmetic() is True or False.
p, n = q.is_prime_power(get_data=True)
if n > 1:
base_ring = GF(p)
elif n == 0:
raise ArithmeticError("q must be a prime power")
else:
raise ValueError("The size of the finite field must not be prime.")
FiniteField_generic.__init__(self, base_ring, name, normalize=True)
self._kwargs = {}
self.__char = p
self.__pari_one = pari.pari(1).Mod(self.__char)
self.__degree = n
self.__order = q
self.__is_field = True
if not sage.rings.polynomial.polynomial_element.is_Polynomial(modulus):
from sage.misc.superseded import deprecation
deprecation(16930, "constructing a FiniteField_ext_pari without giving a polynomial as modulus is deprecated, use the more general FiniteField constructor instead")
if modulus is None or modulus == "default":
from conway_polynomials import exists_conway_polynomial
if exists_conway_polynomial(self.__char, self.__degree):
modulus = "conway"
else:
modulus = "random"
if isinstance(modulus,str):
if modulus == "conway":
from conway_polynomials import conway_polynomial
modulus = conway_polynomial(self.__char, self.__degree)
elif modulus == "random":
# The following is fast/deterministic, but has serious problems since
# it crashes on 64-bit machines, and I can't figure out why:
# self.__pari_modulus = pari.pari.finitefield_init(self.__char, self.__degree, self.variable_name())
# So instead we iterate through random polys until we find an irreducible one.
R = GF(self.__char)['x']
while True:
modulus = R.random_element(self.__degree)
modulus = modulus.monic()
if modulus.degree() == self.__degree and modulus.is_irreducible():
break
else:
raise ValueError("Modulus parameter not understood")
elif isinstance(modulus, (list, tuple)):
modulus = GF(self.__char)['x'](modulus)
elif sage.rings.polynomial.polynomial_element.is_Polynomial(modulus):
if modulus.base_ring() is not base_ring:
modulus = modulus.change_ring(base_ring)
else:
raise ValueError("Modulus parameter not understood")
self._modulus = modulus
f = pari.pari(str(modulus))
self.__pari_modulus = f.subst(modulus.parent().variable_name(), 'a') * self.__pari_one
self.__gen = element_ext_pari.FiniteField_ext_pariElement(self, pari.pari('a'))
self._zero_element = self._element_constructor_(0)
self._one_element = self._element_constructor_(1)
def __init__(self, q, name, modulus=None):
"""
Create finite field of order q with variable printed as name.
INPUT:
- ``q`` -- integer, size of the finite field, not prime
- ``name`` -- variable used for printing element of the finite
field. Also, two finite fields are considered
equal if they have the same variable name, and not
otherwise.
- ``modulus`` -- you may provide a polynomial to use for reduction or
a string:
'conway': force the use of a Conway polynomial, will
raise a RuntimeError if none is found in the database;
'random': use a random irreducible polynomial.
'default': a Conway polynomial is used if found. Otherwise
a random polynomial is used.
OUTPUT:
- FiniteField_ext_pari -- a finite field of order q with given variable name.
EXAMPLES::
sage: FiniteField(65537)
Finite Field of size 65537
sage: FiniteField(2^20, 'c')
Finite Field in c of size 2^20
sage: FiniteField(3^11, "b")
Finite Field in b of size 3^11
sage: FiniteField(3^11, "b").gen()
b
You can also create a finite field using GF, which is a synonym
for FiniteField. ::
sage: GF(19^5, 'a')
Finite Field in a of size 19^5
"""
if element_ext_pari.dynamic_FiniteField_ext_pariElement is None:
element_ext_pari._late_import()
from constructor import FiniteField as GF
q = integer.Integer(q)
if q < 2:
raise ArithmeticError, "q must be a prime power"
from sage.structure.proof.all import arithmetic
proof = arithmetic()
if proof:
F = q.factor()
else:
from sage.rings.arith import is_pseudoprime_small_power
F = is_pseudoprime_small_power(q, get_data=True)
if len(F) != 1:
raise ArithmeticError, "q must be a prime power"
if F[0][1] > 1:
base_ring = GF(F[0][0])
else:
raise ValueError, "The size of the finite field must not be prime."
#base_ring = self
FiniteField_generic.__init__(self, base_ring, name, normalize=True)
self._kwargs = {}
self.__char = F[0][0]
self.__pari_one = pari.pari(1).Mod(self.__char)
self.__degree = integer.Integer(F[0][1])
self.__order = q
self.__is_field = True
if modulus is None or modulus == "default":
from constructor import exists_conway_polynomial
if exists_conway_polynomial(self.__char, self.__degree):
modulus = "conway"
else:
modulus = "random"
if isinstance(modulus, str):
if modulus == "conway":
from constructor import conway_polynomial
modulus = conway_polynomial(self.__char, self.__degree)
elif modulus == "random":
# The following is fast/deterministic, but has serious problems since
# it crashes on 64-bit machines, and I can't figure out why:
# self.__pari_modulus = pari.pari.finitefield_init(self.__char, self.__degree, self.variable_name())
# So instead we iterate through random polys until we find an irreducible one.
R = GF(self.__char)['x']
while True:
modulus = R.random_element(self.__degree)
modulus = modulus.monic()
if modulus.degree(
) == self.__degree and modulus.is_irreducible():
break
else:
raise ValueError("Modulus parameter not understood")
elif isinstance(modulus, (list, tuple)):
modulus = GF(self.__char)['x'](modulus)
elif sage.rings.polynomial.polynomial_element.is_Polynomial(modulus):
if modulus.parent() is not base_ring:
modulus = modulus.change_ring(base_ring)
else:
raise ValueError("Modulus parameter not understood")
self.__modulus = modulus
f = pari.pari(str(modulus))
self.__pari_modulus = f.subst(modulus.parent().variable_name(),
'a') * self.__pari_one
self.__gen = element_ext_pari.FiniteField_ext_pariElement(
self, pari.pari('a'))
self._zero_element = self._element_constructor_(0)
self._one_element = self._element_constructor_(1)
