# Finite Fields of Characteristic 2¶

TESTS:

Test backwards compatibility:

sage: from sage.rings.finite_rings.finite_field_ntl_gf2e import FiniteField_ntl_gf2e
sage: FiniteField_ntl_gf2e(16, 'a')
doctest:...: DeprecationWarning: constructing a FiniteField_ntl_gf2e without giving a polynomial as modulus is deprecated, use the more general FiniteField constructor instead
See http://trac.sagemath.org/16983 for details.
Finite Field in a of size 2^4

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

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

characteristic()

Return the characteristic of self which is 2.

EXAMPLES:

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

degree()

If this field has cardinality $$2^n$$ this method returns $$n$$.

EXAMPLES:

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

fetch_int(number)

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.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]

gen(n=0)

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


TESTS:

sage: GF(2, impl='ntl').gen()
1
sage: GF(2, impl='ntl', modulus=polygen(GF(2)) ).gen()
0
sage: GF(2^19, 'a').gen(1)
Traceback (most recent call last):
...
IndexError: only one generator

order()

Return the cardinality of this field.

EXAMPLES:

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

prime_subfield()

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

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


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