Givaro finite fields

Finite fields that are implemented using Zech logs and the cardinality must be less than \(2^{16}\). By default, Conway polynomials are used as minimal polynomial.

class sage.rings.finite_rings.finite_field_givaro.FiniteField_givaro(q, name='a', modulus=None, repr='poly', cache=False)

Bases: FiniteField

Finite field implemented using Zech logs and the cardinality must be less than \(2^{16}\). By default, Conway polynomials are used as minimal polynomials.

INPUT:

• q – \(p^n\) (must be prime power)

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

• modulus – A minimal polynomial to use for reduction.

• repr – (default: 'poly') controls the way elements are printed to the user:

  • ‘log’: repr is log_repr()

  • ‘int’: repr is int_repr()

  • ‘poly’: repr is poly_repr()

• cache – (default: False) if True a cache of all elements of this field is created. Thus, arithmetic does not create new elements which speeds calculations up. Also, if many elements are needed during a calculation this cache reduces the memory requirement as at most order() elements are created.

OUTPUT:

Givaro finite field with characteristic \(p\) and cardinality \(p^n\).

EXAMPLES:

By default, Conway polynomials are used for extension fields:

sage: k.<a> = GF(2**8)
sage: -a ^ k.degree()
a^4 + a^3 + a^2 + 1
sage: f = k.modulus(); f
x^8 + x^4 + x^3 + x^2 + 1

You may enforce a modulus:

sage: P.<x> = PolynomialRing(GF(2))
sage: f = x^8 + x^4 + x^3 + x + 1 # Rijndael Polynomial
sage: k.<a> = GF(2^8, modulus=f)
sage: k.modulus()
x^8 + x^4 + x^3 + x + 1
sage: a^(2^8)
a

You may enforce a random modulus:

sage: k = GF(3**5, 'a', modulus='random')
sage: k.modulus() # random polynomial
x^5 + 2*x^4 + 2*x^3 + x^2 + 2

Three different representations are possible:

sage: FiniteField(9, 'a', impl='givaro', repr='poly').gen()
a
sage: FiniteField(9, 'a', impl='givaro', repr='int').gen()
3
sage: FiniteField(9, 'a', impl='givaro', repr='log').gen()
1

For prime fields, the default modulus is the polynomial \(x - 1\), but you can ask for a different modulus:

sage: GF(1009, impl='givaro').modulus()
x + 1008
sage: GF(1009, impl='givaro', modulus='conway').modulus()
x + 998

a_times_b_minus_c(a, b, c)

Return a*b - c.

INPUT:

• a,b,c – FiniteField_givaroElement

EXAMPLES:

sage: k.<a> = GF(3**3)
sage: k.a_times_b_minus_c(a,a,k(1))
a^2 + 2

a_times_b_plus_c(a, b, c)

Return a*b + c. This is faster than multiplying a and b first and adding c to the result.

INPUT:

• a,b,c – FiniteField_givaroElement

EXAMPLES:

sage: k.<a> = GF(2**8)
sage: k.a_times_b_plus_c(a,a,k(1))
a^2 + 1

c_minus_a_times_b(a, b, c)

Return c - a*b.

INPUT:

• a,b,c – FiniteField_givaroElement

EXAMPLES:

sage: k.<a> = GF(3**3)
sage: k.c_minus_a_times_b(a,a,k(1))
2*a^2 + 1

characteristic()

Return the characteristic of this field.

EXAMPLES:

sage: p = GF(19^5,'a').characteristic(); p
19
sage: type(p)
<class 'sage.rings.integer.Integer'>

degree()

If the cardinality of self is \(p^n\), then this returns \(n\).

OUTPUT:

Integer – the degree

EXAMPLES:

sage: GF(3^4,'a').degree()
4

fetch_int(*args, **kwds)

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

frobenius_endomorphism(n=1)

INPUT:

• n – an integer (default: 1)

OUTPUT:

The \(n\)-th power of the absolute arithmetic Frobenius endomorphism on this finite field.

EXAMPLES:

sage: k.<t> = GF(3^5)
sage: Frob = k.frobenius_endomorphism(); Frob
Frobenius endomorphism t |--> t^3 on Finite Field in t of size 3^5

sage: a = k.random_element()
sage: Frob(a) == a^3
True

We can specify a power:

sage: k.frobenius_endomorphism(2)
Frobenius endomorphism t |--> t^(3^2) on Finite Field in t of size 3^5

The result is simplified if possible:

sage: k.frobenius_endomorphism(6)
Frobenius endomorphism t |--> t^3 on Finite Field in t of size 3^5
sage: k.frobenius_endomorphism(5)
Identity endomorphism of Finite Field in t of size 3^5

Comparisons work:

sage: k.frobenius_endomorphism(6) == Frob
True
sage: from sage.categories.morphism import IdentityMorphism
sage: k.frobenius_endomorphism(5) == IdentityMorphism(k)
True

AUTHOR:

• Xavier Caruso (2012-06-29)

from_integer(n)

Given an integer \(n\) return a finite field element in self which equals \(n\) under the condition that gen() is set to characteristic().

EXAMPLES:

sage: k.<a> = GF(2^8)
sage: k.from_integer(8)
a^3
sage: e = k.from_integer(151); e
a^7 + a^4 + a^2 + a + 1
sage: 2^7 + 2^4 + 2^2 + 2 + 1
151

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 = GF(3^4, 'b'); k.gen()
b
sage: k.gen(1)
Traceback (most recent call last):
...
IndexError: only one generator
sage: F = FiniteField(31, impl='givaro')
sage: F.gen()
1

int_to_log(n)

Given an integer \(n\) this method returns \(i\) where \(i\) satisfies \(g^i = n \mod p\) where \(g\) is the generator and \(p\) is the characteristic of self.

INPUT:

• n – integer representation of a finite field element

OUTPUT:

log representation of n

EXAMPLES:

sage: k = GF(7**3, 'a')
sage: k.int_to_log(4)
228
sage: k.int_to_log(3)
57
sage: k.gen()^57
3

log_to_int(n)

Given an integer \(n\) this method returns i where i satisfies \(g^n = i\) where \(g\) is the generator of self; the result is interpreted as an integer.

INPUT:

• n – log representation of a finite field element

OUTPUT:

integer representation of a finite field element.

EXAMPLES:

sage: k = GF(2**8, 'a')
sage: k.log_to_int(4)
16
sage: k.log_to_int(20)
180

order()

Return the cardinality of this field.

OUTPUT:

Integer – the number of elements in self.

EXAMPLES:

sage: n = GF(19^5,'a').order(); n
2476099
sage: type(n)
<class 'sage.rings.integer.Integer'>

prime_subfield()

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

EXAMPLES:

sage: GF(3^4, 'b').prime_subfield()
Finite Field of size 3

sage: S.<b> = GF(5^2); S
Finite Field in b of size 5^2
sage: S.prime_subfield()
Finite Field of size 5
sage: type(S.prime_subfield())
<class 'sage.rings.finite_rings.finite_field_prime_modn.FiniteField_prime_modn_with_category'>

random_element(*args, **kwds)

Return a random element of self.

EXAMPLES:

sage: k = GF(23**3, 'a')
sage: e = k.random_element()
sage: e.parent() is k
True
sage: type(e)
<class 'sage.rings.finite_rings.element_givaro.FiniteField_givaroElement'>

sage: P.<x> = PowerSeriesRing(GF(3^3, 'a'))
sage: P.random_element(5).parent() is P
True