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# Givaro Finite Field

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.

TESTS:

Test backwards compatibility:

    sage: from sage.rings.finite_rings.finite_field_givaro import FiniteField_givaro
    sage: FiniteField_givaro(9, 'a')
    doctest:...: DeprecationWarning: constructing a FiniteField_givaro without giving a polynomial as modulus is deprecated, use the more general FiniteField constructor instead
    See http://trac.sagemath.org/16930 for details.
    Finite Field in a of size 3^2

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

  Bases: `sage.rings.finite_rings.finite_field_base.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)
        <type '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`(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.fetch_int(8)
        a^3
        sage: e = k.fetch_int(151); e
        a^7 + a^4 + a^2 + a + 1
        sage: 2^7 + 2^4 + 2^2 + 2 + 1
        151

  `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)
  `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 an 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)
        <type '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(); e
        2*a^2 + 14*a + 21
        sage: type(e)
        <type 'sage.rings.finite_rings.element_givaro.FiniteField_givaroElement'>

        sage: P.<x> = PowerSeriesRing(GF(3^3, 'a'))
        sage: P.random_element(5)
        2*a + 2 + (a^2 + a + 2)*x + (2*a + 1)*x^2 + (2*a^2 + a)*x^3 + 2*a^2*x^4 + O(x^5)

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