Finite Fields of Characteristic 2¶
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class
sage.rings.finite_rings.finite_field_ntl_gf2e.
FiniteField_ntl_gf2e
(q, names='a', modulus=None, repr='poly')¶ Bases:
sage.rings.finite_rings.finite_field_base.FiniteField
Finite Field of characteristic 2 and order 2n.
INPUT:
q
– 2n (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 2n.
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
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characteristic
()¶ Return the characteristic of
self
which is 2.EXAMPLES:
sage: k.<a> = GF(2^16,modulus='random') sage: k.characteristic() 2
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degree
()¶ If this field has cardinality 2n this method returns n.
EXAMPLES:
sage: k.<a> = GF(2^64) sage: k.degree() 64
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fetch_int
(number)¶ Given an integer n less than
cardinality()
with base 2 representation a0+2⋅a1+⋯+2kak, returns a0+a1⋅x+⋯+akxk, where x is the generator of this finite field.INPUT:
number
– an integer
EXAMPLES:
sage: k.<a> = GF(2^48) sage: k.fetch_int(2^43 + 2^15 + 1) a^43 + a^15 + 1 sage: k.fetch_int(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]
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gen
(n=0)¶ Return a generator of
self
over its prime field, which is a root ofself.modulus()
.INPUT:
n
– must be 0
OUTPUT:
An element a of
self
such thatself.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 themodulus="primitive"
option when constructing the field.EXAMPLES:
sage: k.<a> = GF(2^19) sage: k.gen() == a True sage: a a
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order
()¶ Return the cardinality of this field.
EXAMPLES:
sage: k.<a> = GF(2^64) sage: k.order() 18446744073709551616
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prime_subfield
()¶ Return the prime subfield Fp of
self
ifself
is Fpn.EXAMPLES:
sage: F.<a> = GF(2^16) sage: F.prime_subfield() Finite Field of size 2
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sage.rings.finite_rings.finite_field_ntl_gf2e.
late_import
()¶ 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