# Finite Fields¶

Sage supports arithmetic in finite prime and extension fields. Several implementation for prime fields are implemented natively in Sage for several sizes of primes $$p$$. These implementations are

• sage.rings.finite_rings.integer_mod.IntegerMod_int,
• sage.rings.finite_rings.integer_mod.IntegerMod_int64, and
• sage.rings.finite_rings.integer_mod.IntegerMod_gmp.

Small extension fields of cardinality $$< 2^{16}$$ are implemented using tables of Zech logs via the Givaro C++ library (sage.rings.finite_rings.finite_field_givaro.FiniteField_givaro). While this representation is very fast it is limited to finite fields of small cardinality. Larger finite extension fields of order $$q >= 2^{16}$$ are internally represented as polynomials over smaller finite prime fields. If the characteristic of such a field is 2 then NTL is used internally to represent the field (sage.rings.finite_rings.finite_field_ntl_gf2e.FiniteField_ntl_gf2e). In all other case the PARI C library is used (sage.rings.finite_rings.finite_field_pari_ffelt.FiniteField_pari_ffelt).

However, this distinction is internal only and the user usually does not have to worry about it because consistency across all implementations is aimed for. In all extension field implementations the user may either specify a minimal polynomial or leave the choice to Sage.

For small finite fields the default choice are Conway polynomials.

The Conway polynomial $$C_n$$ is the lexicographically first monic irreducible, primitive polynomial of degree $$n$$ over $$GF(p)$$ with the property that for a root $$\alpha$$ of $$C_n$$ we have that $$\beta= \alpha^{(p^n - 1)/(p^m - 1)}$$ is a root of $$C_m$$ for all $$m$$ dividing $$n$$. Sage contains a database of Conway polynomials which also can be queried independently of finite field construction.

A pseudo-Conway polynomial satisfies all of the conditions required of a Conway polynomial except the condition that it is lexicographically first. They are therefore not unique. If no variable name is specified for an extension field, Sage will fit the finite field into a compatible lattice of field extensions defined by pseudo-Conway polynomials. This lattice is stored in an AlgebraicClosureFiniteField object; different algebraic closure objects can be created by using a different prefix keyword to the finite field constructor.

Note that the computation of pseudo-Conway polynomials is expensive when the degree is large and highly composite. If a variable name is specified then a random polynomial is used instead, which will be much faster to find.

While Sage supports basic arithmetic in finite fields some more advanced features for computing with finite fields are still not implemented. For instance, Sage does not calculate embeddings of finite fields yet.

EXAMPLES:

sage: k = GF(5); type(k)
<class 'sage.rings.finite_rings.finite_field_prime_modn.FiniteField_prime_modn_with_category'>

sage: k = GF(5^2,'c'); type(k)
<class 'sage.rings.finite_rings.finite_field_givaro.FiniteField_givaro_with_category'>

sage: k = GF(2^16,'c'); type(k)
<class 'sage.rings.finite_rings.finite_field_ntl_gf2e.FiniteField_ntl_gf2e_with_category'>

sage: k = GF(3^16,'c'); type(k)
<class 'sage.rings.finite_rings.finite_field_pari_ffelt.FiniteField_pari_ffelt_with_category'>


Finite Fields support iteration, starting with 0.

sage: k = GF(9, 'a')
sage: for i,x in enumerate(k):  print("{} {}".format(i, x))
0 0
1 a
2 a + 1
3 2*a + 1
4 2
5 2*a
6 2*a + 2
7 a + 2
8 1
sage: for a in GF(5):
....:     print(a)
0
1
2
3
4


We output the base rings of several finite fields.

sage: k = GF(3); type(k)
<class 'sage.rings.finite_rings.finite_field_prime_modn.FiniteField_prime_modn_with_category'>
sage: k.base_ring()
Finite Field of size 3

sage: k = GF(9,'alpha'); type(k)
<class 'sage.rings.finite_rings.finite_field_givaro.FiniteField_givaro_with_category'>
sage: k.base_ring()
Finite Field of size 3

sage: k = GF(3^40,'b'); type(k)
<class 'sage.rings.finite_rings.finite_field_pari_ffelt.FiniteField_pari_ffelt_with_category'>
sage: k.base_ring()
Finite Field of size 3


Further examples:

sage: GF(2).is_field()
True
sage: GF(next_prime(10^20)).is_field()
True
sage: GF(19^20,'a').is_field()
True
sage: GF(8,'a').is_field()
True


AUTHORS:

• William Stein: initial version
• Robert Bradshaw: prime field implementation
• Martin Albrecht: Givaro and ntl.GF2E implementations
class sage.rings.finite_rings.finite_field_constructor.FiniteFieldFactory

Return the globally unique finite field of given order with generator labeled by the given name and possibly with given modulus.

INPUT:

• order – a prime power
• name – string, optional. Note that there can be a substantial speed penalty (in creating extension fields) when omitting the variable name, since doing so triggers the computation of pseudo-Conway polynomials in order to define a coherent lattice of extensions of the prime field. The speed penalty grows with the size of extension degree and with the number of factors of the extension degree.
• modulus – (optional) either a defining polynomial for the field, or a string specifying an algorithm to use to generate such a polynomial. If modulus is a string, it is passed to irreducible_element() as the parameter algorithm; see there for the permissible values of this parameter. In particular, you can specify modulus="primitive" to get a primitive polynomial. You may not specify a modulus if you do not specify a variable name.
• impl – (optional) a string specifying the implementation of the finite field. Possible values are:
• 'modn' – ring of integers modulo $$p$$ (only for prime fields).
• 'givaro' – Givaro, which uses Zech logs (only for fields of at most 65521 elements).
• 'ntl' – NTL using GF2X (only in characteristic 2).
• 'pari' or 'pari_ffelt' – PARI’s FFELT type (only for extension fields).
• elem_cache – (default: order < 500) cache all elements to avoid creation time; ignored unless impl='givaro'
• repr – (default: 'poly') ignored unless impl='givaro'; controls the way elements are printed to the user:
• ‘log’: repr is log_repr()
• ‘int’: repr is int_repr()
• ‘poly’: repr is poly_repr()
• check_irreducible – verify that the polynomial modulus is irreducible
• proof – bool (default: True): if True, use provable primality test; otherwise only use pseudoprimality test.

ALIAS: You can also use GF instead of FiniteField – they are identical.

EXAMPLES:

sage: k.<a> = FiniteField(9); k
Finite Field in a of size 3^2
sage: parent(a)
Finite Field in a of size 3^2
sage: charpoly(a, 'y')
y^2 + 2*y + 2


We illustrate the proof flag. The following example would hang for a very long time if we didn’t use proof=False.

Note

Magma only supports proof=False for making finite fields, so falsely appears to be faster than Sage – see trac ticket #10975.

sage: k = FiniteField(10^1000 + 453, proof=False)
sage: k = FiniteField((10^1000 + 453)^2, 'a', proof=False)      # long time -- about 5 seconds

sage: F.<x> = GF(5)[]
sage: K.<a> = GF(5**5, name='a', modulus=x^5 - x +1 )
sage: f = K.modulus(); f
x^5 + 4*x + 1
sage: type(f)
<type 'sage.rings.polynomial.polynomial_zmod_flint.Polynomial_zmod_flint'>


By default, the given generator is not guaranteed to be primitive (a generator of the multiplicative group), use modulus="primitive" if you need this:

sage: K.<a> = GF(5^40)
sage: a.multiplicative_order()
189478062869360049565633138
sage: a.is_square()
True
sage: K.<b> = GF(5^40, modulus="primitive")
sage: b.multiplicative_order()
9094947017729282379150390624


The modulus must be irreducible:

sage: K.<a> = GF(5**5, name='a', modulus=x^5 - x)
Traceback (most recent call last):
...
ValueError: finite field modulus must be irreducible but it is not


You can’t accidentally fool the constructor into thinking the modulus is irreducible when it is not, since it actually tests irreducibility modulo $$p$$. Also, the modulus has to be of the right degree (this is always checked):

sage: F.<x> = QQ[]
sage: factor(x^5 + 2)
x^5 + 2
sage: K.<a> = GF(5^5, modulus=x^5 + 2)
Traceback (most recent call last):
...
ValueError: finite field modulus must be irreducible but it is not
sage: K.<a> = GF(5^5, modulus=x^3 + 3*x + 3, check_irreducible=False)
Traceback (most recent call last):
...
ValueError: the degree of the modulus does not equal the degree of the field


Any type which can be converted to the polynomial ring $$GF(p)[x]$$ is accepted as modulus:

sage: K.<a> = GF(13^3, modulus=[1,0,0,2])
sage: K.<a> = GF(13^10, modulus=pari("ffinit(13,10)"))
sage: var('x')
x
sage: K.<a> = GF(13^2, modulus=x^2 - 2)
sage: K.<a> = GF(13^2, modulus=sin(x))
Traceback (most recent call last):
...
TypeError: self must be a numeric expression


If you wish to live dangerously, you can tell the constructor not to test irreducibility using check_irreducible=False, but this can easily lead to crashes and hangs – so do not do it unless you know that the modulus really is irreducible!

sage: K.<a> = GF(5**2, name='a', modulus=x^2 + 2, check_irreducible=False)


Even for prime fields, you can specify a modulus. This will not change how Sage computes in this field, but it will change the result of the modulus() and gen() methods:

sage: k.<a> = GF(5, modulus="primitive")
sage: k.modulus()
x + 3
sage: a
2


The order of a finite field must be a prime power:

sage: GF(1)
Traceback (most recent call last):
...
ValueError: the order of a finite field must be at least 2
sage: GF(100)
Traceback (most recent call last):
...
ValueError: the order of a finite field must be a prime power


Finite fields with explicit random modulus are not cached:

sage: k.<a> = GF(5**10, modulus='random')
sage: n.<a> = GF(5**10, modulus='random')
sage: n is k
False
sage: GF(5**10, 'a') is GF(5**10, 'a')
True


We check that various ways of creating the same finite field yield the same object, which is cached:

sage: K = GF(7, 'a')
sage: L = GF(7, 'b')
sage: K is L           # name is ignored for prime fields
True
sage: K is GF(7, modulus=K.modulus())
True
sage: K = GF(4,'a'); K.modulus()
x^2 + x + 1
sage: L = GF(4,'a', K.modulus())
sage: K is L
True
sage: M = GF(4,'a', K.modulus().change_variable_name('y'))
sage: K is M
True


You may print finite field elements as integers. This currently only works if the order of field is $$<2^{16}$$, though:

sage: k.<a> = GF(2^8, repr='int')
sage: a
2


The following demonstrate coercions for finite fields using Conway polynomials:

sage: k = GF(5^2); a = k.gen()
sage: l = GF(5^5); b = l.gen()
sage: a + b
3*z10^5 + z10^4 + z10^2 + 3*z10 + 1


Note that embeddings are compatible in lattices of such finite fields:

sage: m = GF(5^3); c = m.gen()
sage: (a+b)+c == a+(b+c)
True
sage: (a*b)*c == a*(b*c)
True
sage: from sage.categories.pushout import pushout
sage: n = pushout(k, l)
sage: o = pushout(l, m)
sage: q = pushout(n, o)
sage: q(o(b)) == q(n(b))
True


Another check that embeddings are defined properly:

sage: k = GF(3**10)
sage: l = GF(3**20)
sage: l(k.gen()**10) == l(k.gen())**10
True


Using pseudo-Conway polynomials is slow for highly composite extension degrees:

sage: k = GF(3^120) # long time -- about 3 seconds
sage: GF(3^40).gen().minimal_polynomial()(k.gen()^((3^120-1)/(3^40-1))) # long time because of previous line
0


Before trac ticket #17569, the boolean keyword argument conway was required when creating finite fields without a variable name. This keyword argument is now removed (trac ticket #21433). You can still pass in prefix as an argument, which has the effect of changing the variable name of the algebraic closure:

sage: K = GF(3^10, prefix='w'); L = GF(3^10); K is L
False
sage: K.variable_name(), L.variable_name()
('w10', 'z10')
sage: list(K.polynomial()) == list(L.polynomial())
True

create_key_and_extra_args(order, name=None, modulus=None, names=None, impl=None, proof=None, check_irreducible=True, prefix=None, repr=None, elem_cache=None, **kwds)

EXAMPLES:

sage: GF.create_key_and_extra_args(9, 'a')
((9, ('a',), x^2 + 2*x + 2, 'givaro', 3, 2, True, None, 'poly', True), {})


We do not take invalid keyword arguments and raise a value error to better ensure uniqueness:

sage: GF.create_key_and_extra_args(9, 'a', foo='value')
Traceback (most recent call last):
...
TypeError: create_key_and_extra_args() got an unexpected keyword argument 'foo'


Moreover, repr and elem_cache are ignored when not using givaro:

sage: GF.create_key_and_extra_args(16, 'a', impl='ntl', repr='poly')
((16, ('a',), x^4 + x + 1, 'ntl', 2, 4, True, None, None, None), {})
sage: GF.create_key_and_extra_args(16, 'a', impl='ntl', elem_cache=False)
((16, ('a',), x^4 + x + 1, 'ntl', 2, 4, True, None, None, None), {})
sage: GF(16, impl='ntl') is GF(16, impl='ntl', repr='foo')
True


We handle extra arguments for the givaro finite field and create unique objects for their defaults:

sage: GF(25, impl='givaro') is GF(25, impl='givaro', repr='poly')
True
sage: GF(25, impl='givaro') is GF(25, impl='givaro', elem_cache=True)
True
sage: GF(625, impl='givaro') is GF(625, impl='givaro', elem_cache=False)
True


We explicitly take structure, implementation and prec attributes for compatibility with AlgebraicExtensionFunctor but we ignore them as they are not used, see trac ticket #21433:

sage: GF.create_key_and_extra_args(9, 'a', structure=None)
((9, ('a',), x^2 + 2*x + 2, 'givaro', 3, 2, True, None, 'poly', True), {})

create_object(version, key, **kwds)

EXAMPLES:

sage: K = GF(19) # indirect doctest
sage: TestSuite(K).run()


We try to create finite fields with various implementations:

sage: k = GF(2, impl='modn')
sage: k = GF(2, impl='givaro')
sage: k = GF(2, impl='ntl')
sage: k = GF(2, impl='pari')
Traceback (most recent call last):
...
ValueError: the degree must be at least 2
sage: k = GF(2, impl='supercalifragilisticexpialidocious')
Traceback (most recent call last):
...
ValueError: no such finite field implementation: 'supercalifragilisticexpialidocious'
sage: k.<a> = GF(2^15, impl='modn')
Traceback (most recent call last):
...
ValueError: the 'modn' implementation requires a prime order
sage: k.<a> = GF(2^15, impl='givaro')
sage: k.<a> = GF(2^15, impl='ntl')
sage: k.<a> = GF(2^15, impl='pari')
sage: k.<a> = GF(3^60, impl='modn')
Traceback (most recent call last):
...
ValueError: the 'modn' implementation requires a prime order
sage: k.<a> = GF(3^60, impl='givaro')
Traceback (most recent call last):
...
ValueError: q must be < 2^16
sage: k.<a> = GF(3^60, impl='ntl')
Traceback (most recent call last):
...
ValueError: q must be a 2-power
sage: k.<a> = GF(3^60, impl='pari')

sage.rings.finite_rings.finite_field_constructor.is_PrimeFiniteField(x)

Returns True if x is a prime finite field.

EXAMPLES:

sage: from sage.rings.finite_rings.finite_field_constructor import is_PrimeFiniteField
sage: is_PrimeFiniteField(QQ)
False
sage: is_PrimeFiniteField(GF(7))
True
sage: is_PrimeFiniteField(GF(7^2,'a'))
False
sage: is_PrimeFiniteField(GF(next_prime(10^90,proof=False)))
True