Finite fields of characteristic 2#

class sage.rings.finite_rings.finite_field_ntl_gf2e.FiniteField_ntl_gf2e(q, names='a', modulus=None, repr='poly')[source]#

Bases: FiniteField

Finite Field of characteristic 2 and order \(2^n\).

INPUT:

  • q – \(2^n\) (must be 2 power)

  • names – variable used for poly_repr (default: 'a')

  • modulus – A minimal polynomial to use for reduction.

  • 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(6397)
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(2^23, modulus='conway')
sage: a.multiplicative_order() == k.order() - 1
True

>>> from sage.all import *
>>> k = GF(Integer(2)**Integer(16), names=('a',)); (a,) = k._first_ngens(1)
>>> type(k)
<class 'sage.rings.finite_rings.finite_field_ntl_gf2e.FiniteField_ntl_gf2e_with_category'>
>>> k = GF(Integer(2)**Integer(1024), names=('a',)); (a,) = k._first_ngens(1)
>>> k.modulus()
x^1024 + x^19 + x^6 + x + 1
>>> set_random_seed(Integer(6397))
>>> k = GF(Integer(2)**Integer(17), modulus='random', names=('a',)); (a,) = k._first_ngens(1)
>>> k.modulus()
x^17 + x^16 + x^15 + x^10 + x^8 + x^6 + x^4 + x^3 + x^2 + x + 1
>>> k.modulus().is_irreducible()
True
>>> k = GF(Integer(2)**Integer(211), modulus='minimal_weight', names=('a',)); (a,) = k._first_ngens(1)
>>> k.modulus()
x^211 + x^11 + x^10 + x^8 + 1
>>> k = GF(Integer(2)**Integer(211), modulus='conway', names=('a',)); (a,) = k._first_ngens(1)
>>> k.modulus()
x^211 + x^9 + x^6 + x^5 + x^3 + x + 1
>>> k = GF(Integer(2)**Integer(23), modulus='conway', names=('a',)); (a,) = k._first_ngens(1)
>>> a.multiplicative_order() == k.order() - Integer(1)
True
characteristic()[source]#

Return the characteristic of self which is 2.

EXAMPLES:

sage: k.<a> = GF(2^16,modulus='random')
sage: k.characteristic()
2

>>> from sage.all import *
>>> k = GF(Integer(2)**Integer(16),modulus='random', names=('a',)); (a,) = k._first_ngens(1)
>>> k.characteristic()
2
degree()[source]#

If this field has cardinality \(2^n\) this method returns \(n\).

EXAMPLES:

sage: k.<a> = GF(2^64)
sage: k.degree()
64

>>> from sage.all import *
>>> k = GF(Integer(2)**Integer(64), names=('a',)); (a,) = k._first_ngens(1)
>>> k.degree()
64
fetch_int(*args, **kwds)[source]#

Deprecated: Use from_integer() instead. See Issue #33941 for details.

from_integer(number)[source]#

Given an integer \(n\) less than cardinality() with base \(2\) representation \(a_0 + 2 \cdot a_1 + \cdots + 2^k a_k\), returns \(a_0 + a_1 \cdot x + \cdots + a_k x^k\), where \(x\) is the generator of this finite field.

INPUT:

  • number – an integer

EXAMPLES:

sage: k.<a> = GF(2^48)
sage: k.from_integer(2^43 + 2^15 + 1)
a^43 + a^15 + 1
sage: k.from_integer(33793)
a^15 + a^10 + 1
sage: 33793.digits(2) # little endian
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1]

>>> from sage.all import *
>>> k = GF(Integer(2)**Integer(48), names=('a',)); (a,) = k._first_ngens(1)
>>> k.from_integer(Integer(2)**Integer(43) + Integer(2)**Integer(15) + Integer(1))
a^43 + a^15 + 1
>>> k.from_integer(Integer(33793))
a^15 + a^10 + 1
>>> Integer(33793).digits(Integer(2)) # little endian
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1]
gen(n=0)[source]#

Return a generator of self over its prime field, which is a root of self.modulus().

INPUT:

  • n – must be 0

OUTPUT:

An element \(a\) of self such that self.modulus()(a) == 0.

Warning

This generator is not guaranteed to be a generator for the multiplicative group. To obtain the latter, use multiplicative_generator() or use the modulus="primitive" option when constructing the field.

EXAMPLES:

sage: k.<a> = GF(2^19)
sage: k.gen() == a
True
sage: a
a

>>> from sage.all import *
>>> k = GF(Integer(2)**Integer(19), names=('a',)); (a,) = k._first_ngens(1)
>>> k.gen() == a
True
>>> a
a
order()[source]#

Return the cardinality of this field.

EXAMPLES:

sage: k.<a> = GF(2^64)
sage: k.order()
18446744073709551616

>>> from sage.all import *
>>> k = GF(Integer(2)**Integer(64), names=('a',)); (a,) = k._first_ngens(1)
>>> k.order()
18446744073709551616
prime_subfield()[source]#

Return the prime subfield \(\GF{p}\) of self if self is \(\GF{p^n}\).

EXAMPLES:

sage: F.<a> = GF(2^16)
sage: F.prime_subfield()
Finite Field of size 2

>>> from sage.all import *
>>> F = GF(Integer(2)**Integer(16), names=('a',)); (a,) = F._first_ngens(1)
>>> F.prime_subfield()
Finite Field of size 2
sage.rings.finite_rings.finite_field_ntl_gf2e.late_import()[source]#

Imports various modules after startup.

EXAMPLES:

sage: sage.rings.finite_rings.finite_field_ntl_gf2e.late_import()
sage: sage.rings.finite_rings.finite_field_ntl_gf2e.GF2 is None # indirect doctest
False

>>> from sage.all import *
>>> sage.rings.finite_rings.finite_field_ntl_gf2e.late_import()
>>> sage.rings.finite_rings.finite_field_ntl_gf2e.GF2 is None # indirect doctest
False