# 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.

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 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.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; must be specified unless order is prime.
• 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.
• 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_ffelt' – PARI’s FFELT type (only for extension fields).
• 'pari_mod' – Older PARI implementation using POLMODs (slower than 'pari_ffelt', only for extension fields).
• elem_cache – cache all elements to avoid creation time (default: order < 500)
• check_irreducible – verify that the polynomial modulus is irreducible
• proof – bool (default: True): if True, use provable primality test; otherwise only use pseudoprimality test.
• args – additional parameters passed to finite field implementations
• kwds – additional keyword parameters passed to finite field implementations

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()
4547473508864641189575195312
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: unable to convert x (=sin(x)) to an integer


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, conway=True, prefix='z'); a = k.gen()
sage: l = GF(5^5, conway=True, prefix='z'); 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, conway=True, prefix='z'); 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, conway=True, prefix='z')
sage: l = GF(3**20, conway=True, prefix='z')
sage: l(k.gen()**10) == l(k.gen())**10
True


Check that trac ticket #16934 has been fixed:

sage: k1.<a> = GF(17^14, impl="pari_ffelt")
sage: _ = a/2
sage: k2.<a> = GF(17^14, impl="pari_ffelt")
sage: k1 is k2
True

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

EXAMPLES:

sage: GF.create_key_and_extra_args(9, 'a')
((9, ('a',), x^2 + 2*x + 2, 'givaro', '{}', 3, 2, True), {})
sage: GF.create_key_and_extra_args(9, 'a', foo='value')
((9, ('a',), x^2 + 2*x + 2, 'givaro', "{'foo': 'value'}", 3, 2, True), {'foo': 'value'})

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_ffelt')
Traceback (most recent call last):
...
ValueError: the degree must be at least 2
sage: k = GF(2, impl='pari_mod')
Traceback (most recent call last):
...
ValueError: The size of the finite field must not be prime.
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_ffelt')
sage: k.<a> = GF(2^15, impl='pari_mod')
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_ffelt')
sage: k.<a> = GF(3^60, impl='pari_mod')

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

Returns True if x is a prime finite field.

EXAMPLES:

sage: from sage.rings.finite_rings.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


Rational Numbers

#### Next topic

Base class for finite field elements