def __init__(self, q, names="a", modulus=None, repr="poly"):
"""
Initialize ``self``.
TESTS::
sage: k.<a> = GF(2^100, modulus='strangeinput')
Traceback (most recent call last):
...
ValueError: no such algorithm for finding an irreducible polynomial: strangeinput
sage: k.<a> = GF(2^20) ; type(k)
<class 'sage.rings.finite_rings.finite_field_ntl_gf2e.FiniteField_ntl_gf2e_with_category'>
sage: loads(dumps(k)) is k
True
sage: k1.<a> = GF(2^16)
sage: k2.<a> = GF(2^17)
sage: k1 == k2
False
sage: k3.<a> = GF(2^16, impl="pari_ffelt")
sage: k1 == k3
False
sage: TestSuite(k).run()
sage: k.<a> = GF(2^64)
sage: k._repr_option('element_is_atomic')
False
sage: P.<x> = PolynomialRing(k)
sage: (a+1)*x # indirect doctest
(a + 1)*x
"""
late_import()
q = Integer(q)
if q < 2:
raise ValueError("q must be a 2-power")
k = q.exact_log(2)
if q != 1 << k:
raise ValueError("q must be a 2-power")
FiniteField.__init__(self, GF2, names, normalize=True)
self._kwargs = {'repr': repr}
from sage.rings.polynomial.polynomial_element import is_Polynomial
if not is_Polynomial(modulus):
from sage.misc.superseded import deprecation
deprecation(
16983,
"constructing a FiniteField_ntl_gf2e without giving a polynomial as modulus is deprecated, use the more general FiniteField constructor instead"
)
R = GF2['x']
if modulus is None or isinstance(modulus, str):
modulus = R.irreducible_element(k, algorithm=modulus)
else:
modulus = R(modulus)
self._cache = Cache_ntl_gf2e(self, k, modulus)
self._modulus = modulus
def __init__(self, q, names="a", modulus=None, repr="poly"):
"""
Initialize ``self``.
TESTS::
sage: k.<a> = GF(2^100, modulus='strangeinput')
Traceback (most recent call last):
...
ValueError: Modulus parameter not understood
sage: k.<a> = GF(2^20) ; type(k)
<class 'sage.rings.finite_rings.finite_field_ntl_gf2e.FiniteField_ntl_gf2e_with_category'>
sage: loads(dumps(k)) is k
True
sage: k1.<a> = GF(2^16)
sage: k2.<a> = GF(2^17)
sage: k1 == k2
False
sage: k3 = k1._finite_field_ext_pari_()
sage: k1 == k3
False
sage: TestSuite(k).run()
sage: k.<a> = GF(2^64)
sage: k._repr_option('element_is_atomic')
False
sage: P.<x> = PolynomialRing(k)
sage: (a+1)*x # indirect doctest
(a + 1)*x
"""
late_import()
q = Integer(q)
if q < 2:
raise ValueError("q must be a 2-power")
k = q.exact_log(2)
if q != 1 << k:
raise ValueError("q must be a 2-power")
p = Integer(2)
FiniteField.__init__(self, GF(p), names, normalize=True)
self._kwargs = {'repr': repr}
self._is_conway = False
if modulus is None or modulus == 'default':
if exists_conway_polynomial(p, k):
modulus = "conway"
else:
modulus = "minimal_weight"
if modulus == "conway":
modulus = conway_polynomial(p, k)
self._is_conway = True
if is_Polynomial(modulus):
modulus = modulus.list()
self._cache = Cache_ntl_gf2e(self, k, modulus)
self._polynomial = {}
def __init__(self, q, names="a", modulus=None, repr="poly"):
"""
Initialize ``self``.
TESTS::
sage: k.<a> = GF(2^100, modulus='strangeinput')
Traceback (most recent call last):
...
ValueError: Modulus parameter not understood
sage: k.<a> = GF(2^20) ; type(k)
<class 'sage.rings.finite_rings.finite_field_ntl_gf2e.FiniteField_ntl_gf2e_with_category'>
sage: loads(dumps(k)) is k
True
sage: k1.<a> = GF(2^16)
sage: k2.<a> = GF(2^17)
sage: k1 == k2
False
sage: k3 = k1._finite_field_ext_pari_()
sage: k1 == k3
False
sage: TestSuite(k).run()
sage: k.<a> = GF(2^64)
sage: k._repr_option('element_is_atomic')
False
sage: P.<x> = PolynomialRing(k)
sage: (a+1)*x # indirect doctest
(a + 1)*x
"""
late_import()
q = Integer(q)
if q < 2:
raise ValueError("q must be a 2-power")
k = q.exact_log(2)
if q != 1 << k:
raise ValueError("q must be a 2-power")
p = Integer(2)
FiniteField.__init__(self, GF(p), names, normalize=True)
self._kwargs = {'repr':repr}
self._is_conway = False
if modulus is None or modulus == 'default':
if exists_conway_polynomial(p, k):
modulus = "conway"
else:
modulus = "minimal_weight"
if modulus == "conway":
modulus = conway_polynomial(p, k)
self._is_conway = True
if is_Polynomial(modulus):
modulus = modulus.list()
self._cache = Cache_ntl_gf2e(self, k, modulus)
self._polynomial = {}
def __init__(self, q, names="a", modulus=None, repr="poly"):
"""
Initialize ``self``.
TESTS::
sage: k.<a> = GF(2^100, modulus='strangeinput')
Traceback (most recent call last):
...
ValueError: no such algorithm for finding an irreducible polynomial: strangeinput
sage: k.<a> = GF(2^20) ; type(k)
<class 'sage.rings.finite_rings.finite_field_ntl_gf2e.FiniteField_ntl_gf2e_with_category'>
sage: loads(dumps(k)) is k
True
sage: k1.<a> = GF(2^16)
sage: k2.<a> = GF(2^17)
sage: k1 == k2
False
sage: k3.<a> = GF(2^16, impl="pari_ffelt")
sage: k1 == k3
False
sage: TestSuite(k).run()
sage: k.<a> = GF(2^64)
sage: k._repr_option('element_is_atomic')
False
sage: P.<x> = PolynomialRing(k)
sage: (a+1)*x # indirect doctest
(a + 1)*x
"""
late_import()
q = Integer(q)
if q < 2:
raise ValueError("q must be a 2-power")
k = q.exact_log(2)
if q != 1 << k:
raise ValueError("q must be a 2-power")
FiniteField.__init__(self, GF2, names, normalize=True)
self._kwargs = {'repr':repr}
from sage.rings.polynomial.polynomial_element import is_Polynomial
if not is_Polynomial(modulus):
from sage.misc.superseded import deprecation
deprecation(16983, "constructing a FiniteField_ntl_gf2e without giving a polynomial as modulus is deprecated, use the more general FiniteField constructor instead")
R = GF2['x']
if modulus is None or isinstance(modulus, str):
modulus = R.irreducible_element(k, algorithm=modulus)
else:
modulus = R(modulus)
self._cache = Cache_ntl_gf2e(self, k, modulus)
self._modulus = modulus
def __init__(self, q, names="a", modulus=None, repr="poly"):
"""
Initialize ``self``.
TESTS::
sage: k.<a> = GF(2^100, modulus='strangeinput')
Traceback (most recent call last):
...
ValueError: no such algorithm for finding an irreducible polynomial: strangeinput
sage: k.<a> = GF(2^20) ; type(k)
<class 'sage.rings.finite_rings.finite_field_ntl_gf2e.FiniteField_ntl_gf2e_with_category'>
sage: loads(dumps(k)) is k
True
sage: k1.<a> = GF(2^16)
sage: k2.<a> = GF(2^17)
sage: k1 == k2
False
sage: k3.<a> = GF(2^16, impl="pari_ffelt")
sage: k1 == k3
False
sage: TestSuite(k).run()
sage: k.<a> = GF(2^64)
sage: k._repr_option('element_is_atomic')
False
sage: P.<x> = PolynomialRing(k)
sage: (a+1)*x # indirect doctest
(a + 1)*x
"""
late_import()
q = Integer(q)
if q < 2:
raise ValueError("q must be a 2-power")
k = q.exact_log(2)
if q != 1 << k:
raise ValueError("q must be a 2-power")
FiniteField.__init__(self, GF2, names, normalize=True)
from sage.rings.polynomial.polynomial_element import is_Polynomial
if not is_Polynomial(modulus):
raise TypeError("modulus must be a polynomial")
self._cache = Cache_ntl_gf2e(self, k, modulus)
self._modulus = modulus
def __init__(self, q, names="a", modulus=None, repr="poly"):
"""
Initialize ``self``.
TESTS::
sage: k.<a> = GF(2^100, modulus='strangeinput')
Traceback (most recent call last):
...
ValueError: no such algorithm for finding an irreducible polynomial: strangeinput
sage: k.<a> = GF(2^20) ; type(k)
<class 'sage.rings.finite_rings.finite_field_ntl_gf2e.FiniteField_ntl_gf2e_with_category'>
sage: loads(dumps(k)) is k
True
sage: k1.<a> = GF(2^16)
sage: k2.<a> = GF(2^17)
sage: k1 == k2
False
sage: k3.<a> = GF(2^16, impl="pari_ffelt")
sage: k1 == k3
False
sage: TestSuite(k).run()
sage: k.<a> = GF(2^64)
sage: k._repr_option('element_is_atomic')
False
sage: P.<x> = PolynomialRing(k)
sage: (a+1)*x # indirect doctest
(a + 1)*x
"""
late_import()
q = Integer(q)
if q < 2:
raise ValueError("q must be a 2-power")
k = q.exact_log(2)
if q != 1 << k:
raise ValueError("q must be a 2-power")
FiniteField.__init__(self, GF2, names, normalize=True)
from sage.rings.polynomial.polynomial_element import is_Polynomial
if not is_Polynomial(modulus):
raise TypeError("modulus must be a polynomial")
self._cache = Cache_ntl_gf2e(self, k, modulus)
self._modulus = modulus
def xi_degrees(n, p=2, reverse=True):
r"""
Decreasing list of degrees of the xi_i's, starting in degree n.
INPUT:
- `n` - integer
- `p` - prime number, optional (default 2)
- ``reverse`` - bool, optional (default True)
OUTPUT: ``list`` - list of integers
When `p=2`: decreasing list of the degrees of the `\xi_i`'s with
degree at most n.
At odd primes: decreasing list of these degrees, each divided by
`2(p-1)`.
If ``reverse`` is False, then return an increasing list rather
than a decreasing one.
EXAMPLES::
sage: sage.algebras.steenrod.steenrod_algebra_bases.xi_degrees(17)
[15, 7, 3, 1]
sage: sage.algebras.steenrod.steenrod_algebra_bases.xi_degrees(17, reverse=False)
[1, 3, 7, 15]
sage: sage.algebras.steenrod.steenrod_algebra_bases.xi_degrees(17,p=3)
[13, 4, 1]
sage: sage.algebras.steenrod.steenrod_algebra_bases.xi_degrees(400,p=17)
[307, 18, 1]
"""
from sage.rings.integer import Integer
if n <= 0:
return []
N = Integer(n * (p - 1) + 1)
l = [(p**d - 1) // (p - 1) for d in range(1, N.exact_log(p) + 1)]
if reverse:
l.reverse()
return l
def __init__(self, q, names="a", modulus=None, repr="poly"):
"""
Finite Field for characteristic 2 and order >= 2.
INPUT:
q -- 2^n (must be 2 power)
names -- variable used for poly_repr (default: 'a')
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;
'minimal_weight': use a minimal weight polynomial, should
result in faster arithmetic;
'random': use a random irreducible polynomial.
'default':a Conway polynomial is used if found. Otherwise
a sparse polynomial is used.
repr -- controls the way elements are printed to the user:
(default: 'poly')
'poly': polynomial representation
OUTPUT:
Finite field with characteristic 2 and cardinality 2^n.
EXAMPLES::
sage: k.<a> = GF(2^16)
sage: type(k)
<class 'sage.rings.finite_rings.finite_field_ntl_gf2e.FiniteField_ntl_gf2e_with_category'>
sage: k.<a> = GF(2^1024)
sage: k.modulus()
x^1024 + x^19 + x^6 + x + 1
sage: set_random_seed(0)
sage: k.<a> = GF(2^17, modulus='random')
sage: k.modulus()
x^17 + x^16 + x^15 + x^10 + x^8 + x^6 + x^4 + x^3 + x^2 + x + 1
sage: k.modulus().is_irreducible()
True
sage: k.<a> = GF(2^211, modulus='minimal_weight')
sage: k.modulus()
x^211 + x^11 + x^10 + x^8 + 1
sage: k.<a> = GF(2^211, modulus='conway')
sage: k.modulus()
x^211 + x^9 + x^6 + x^5 + x^3 + x + 1
sage: k.<a> = GF(2^411, modulus='conway')
Traceback (most recent call last):
...
RuntimeError: requested conway polynomial not in database.
TESTS::
sage: k.<a> = GF(2^100, modulus='strangeinput')
Traceback (most recent call last):
...
ValueError: Modulus parameter not understood
sage: k.<a> = GF(2^20) ; type(k)
<class 'sage.rings.finite_rings.finite_field_ntl_gf2e.FiniteField_ntl_gf2e_with_category'>
sage: loads(dumps(k)) is k
True
sage: k1.<a> = GF(2^16)
sage: k2.<a> = GF(2^17)
sage: k1 == k2
False
sage: k3 = k1._finite_field_ext_pari_()
sage: k1 == k3
False
"""
late_import()
q = Integer(q)
if q < 2:
raise ValueError("q must be a 2-power")
k = q.exact_log(2)
if q != 1 << k:
raise ValueError("q must be a 2-power")
p = Integer(2)
FiniteField.__init__(self, GF(p), names, normalize=True)
self._kwargs = {'repr': repr}
self._is_conway = False
if modulus is None or modulus == 'default':
if exists_conway_polynomial(p, k):
modulus = "conway"
else:
modulus = "minimal_weight"
if modulus == "conway":
modulus = conway_polynomial(p, k)
self._is_conway = True
if is_Polynomial(modulus):
modulus = modulus.list()
self._cache = Cache_ntl_gf2e(self, k, modulus)
self._polynomial = {}
def __init__(self, q, names="a", modulus=None, repr="poly"):
"""
Finite Field for characteristic 2 and order >= 2.
INPUT:
q -- 2^n (must be 2 power)
names -- variable used for poly_repr (default: 'a')
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;
'minimal_weight': use a minimal weight polynomial, should
result in faster arithmetic;
'random': use a random irreducible polynomial.
'default':a Conway polynomial is used if found. Otherwise
a sparse polynomial is used.
repr -- controls the way elements are printed to the user:
(default: 'poly')
'poly': polynomial representation
OUTPUT:
Finite field with characteristic 2 and cardinality 2^n.
EXAMPLES::
sage: k.<a> = GF(2^16)
sage: type(k)
<class 'sage.rings.finite_rings.finite_field_ntl_gf2e.FiniteField_ntl_gf2e_with_category'>
sage: k.<a> = GF(2^1024)
sage: k.modulus()
x^1024 + x^19 + x^6 + x + 1
sage: set_random_seed(0)
sage: k.<a> = GF(2^17, modulus='random')
sage: k.modulus()
x^17 + x^16 + x^15 + x^10 + x^8 + x^6 + x^4 + x^3 + x^2 + x + 1
sage: k.modulus().is_irreducible()
True
sage: k.<a> = GF(2^211, modulus='minimal_weight')
sage: k.modulus()
x^211 + x^11 + x^10 + x^8 + 1
sage: k.<a> = GF(2^211, modulus='conway')
sage: k.modulus()
x^211 + x^9 + x^6 + x^5 + x^3 + x + 1
sage: k.<a> = GF(2^411, modulus='conway')
Traceback (most recent call last):
...
RuntimeError: requested conway polynomial not in database.
TESTS::
sage: k.<a> = GF(2^100, modulus='strangeinput')
Traceback (most recent call last):
...
ValueError: Modulus parameter not understood
sage: k.<a> = GF(2^20) ; type(k)
<class 'sage.rings.finite_rings.finite_field_ntl_gf2e.FiniteField_ntl_gf2e_with_category'>
sage: loads(dumps(k)) is k
True
sage: k1.<a> = GF(2^16)
sage: k2.<a> = GF(2^17)
sage: k1 == k2
False
sage: k3 = k1._finite_field_ext_pari_()
sage: k1 == k3
False
"""
late_import()
q = Integer(q)
if q < 2:
raise ValueError("q must be a 2-power")
k = q.exact_log(2)
if q != 1 << k:
raise ValueError("q must be a 2-power")
p = Integer(2)
FiniteField.__init__(self, GF(p), names, normalize=True)
self._kwargs = {'repr':repr}
self._is_conway = False
if modulus is None or modulus == 'default':
if exists_conway_polynomial(p, k):
modulus = "conway"
else:
modulus = "minimal_weight"
if modulus == "conway":
modulus = conway_polynomial(p, k)
self._is_conway = True
if is_Polynomial(modulus):
modulus = modulus.list()
self._cache = Cache_ntl_gf2e(self, k, modulus)
self._polynomial = {}