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# Givaro Field Elements

Givaro Field Elements

Sage includes the Givaro finite field library, for highly optimized arithmetic in finite fields.

Note

The arithmetic is performed by the Givaro C++ library which uses Zech logs internally to represent finite field elements. This implementation is the default finite extension field implementation in Sage for the cardinality less than \(2^{16}\), as it is a lot faster than the PARI implementation. Some functionality in this class however is implemented using PARI.

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'>

    sage: n = previous_prime_power(2^16 - 1)
    sage: while is_prime(n):
    ...    n = previous_prime_power(n)
    sage: factor(n)
    251^2
    sage: k = GF(n,'c'); type(k)
    <class 'sage.rings.finite_rings.finite_field_givaro.FiniteField_givaro_with_category'>

AUTHORS:

- Martin Albrecht <malb@informatik.uni-bremen.de> (2006-06-05)
- William Stein (2006-12-07): editing, lots of docs, etc.
- Robert Bradshaw (2007-05-23): is_square/sqrt, pow.

class `sage.rings.finite_rings.element_givaro.``Cache_givaro`

  Bases: `sage.structure.sage_object.SageObject`

  Finite Field.

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

  INPUT:

  - `q` – \(p^n\) (must be prime power)
  - `name` – variable used for poly_repr (default: `'a'`)
  - `modulus` – a polynomial to use as modulus.
  - `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:

      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

  For binary fields, you may ask for a minimal weight polynomial:

      sage: k = GF(2**10, 'a', modulus='minimal_weight')
      sage: k.modulus()
      x^10 + x^3 + 1

  `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._cache.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._cache.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._cache.c_minus_a_times_b(a,a,k(1))
        2*a^2 + 1

  `characteristic`()

    Return the characteristic of this field.

    EXAMPLES:

        sage: p = GF(19^3,'a')._cache.characteristic(); p
        19

  `element_from_data`(e)

    Coerces several data types to `self`.

    INPUT:

    - `e` – data to coerce in.

    EXAMPLES:

        sage: k = GF(3^8, 'a')
        sage: type(k)
        <class 'sage.rings.finite_rings.finite_field_givaro.FiniteField_givaro_with_category'>
        sage: e = k.vector_space().gen(1); e
        (0, 1, 0, 0, 0, 0, 0, 0)
        sage: k(e) #indirect doctest
        a

    TESTS:

    Check coercion of large integers:

        sage: k(-5^13)
        1
        sage: k(2^31)
        2
        sage: k(int(10^19))
        1
        sage: k(2^63)
        2
        sage: k(2^100)
        1
        sage: k(int(2^100))
        1
        sage: k(long(2^100))
        1
        sage: k(-2^100)
        2

    For more examples, see `finite_field_givaro.FiniteField_givaro._element_constructor_`

  `exponent`()

    Returns the degree of this field over \(\GF{p}\).

    EXAMPLES:

        sage: K.<a> = GF(9); K._cache.exponent()
        2

  `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._cache.fetch_int(8)
        a^3
        sage: e = k._cache.fetch_int(151); e
        a^7 + a^4 + a^2 + a + 1
        sage: 2^7 + 2^4 + 2^2 + 2 + 1
        151

  `gen`()

    Returns a generator of the field.

    EXAMPLES:

        sage: K.<a> = GF(625)
        sage: K._cache.gen()
        a

  `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._cache.int_to_log(4)
        228
        sage: k._cache.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._cache.log_to_int(4)
        16
        sage: k._cache.log_to_int(20)
        180

  `order`()

    Returns the order of this field.

    EXAMPLES:

        sage: K.<a> = GF(9)
        sage: K._cache.order()
        9

  `order_c`()

    Returns the order of this field.

    EXAMPLES:

        sage: K.<a> = GF(9)
        sage: K._cache.order_c()
        9

  `random_element`(*args, **kwds)

    Return a random element of `self`.

    EXAMPLES:

        sage: k = GF(23**3, 'a')
        sage: e = k._cache.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)

  `repr`

class `sage.rings.finite_rings.element_givaro.``FiniteField_givaroElement`

  Bases: `sage.rings.finite_rings.element_base.FinitePolyExtElement`

  An element of a (Givaro) finite field.

  `int_repr`()

    Return the string representation of `self` as an int (as returned by `log_to_int()`).

    EXAMPLES:

        sage: k.<b> = GF(5^2); k
        Finite Field in b of size 5^2
        sage: (b+1).int_repr()
        '6'

  `integer_representation`()

    Return the integer representation of `self`. When `self` is in the prime subfield, the integer returned is equal to `self` and not to `log_repr`.

    Elements of this field are represented as ints in as follows: for \(e \in \GF{p}[x]\) with \(e = a_0 + a_1x + a_2x^2 + \cdots\), \(e\) is represented as: \(n= a_0 + a_1 p + a_2 p^2 + \cdots\).

    EXAMPLES:

        sage: k.<b> = GF(5^2); k
        Finite Field in b of size 5^2
        sage: k(4).integer_representation()
        4
        sage: b.integer_representation()
        5
        sage: type(b.integer_representation())
        <type 'int'>

  `is_one`()

    Return `True` if `self == k(1)`.

    EXAMPLES:

        sage: k.<a> = GF(3^4); k
        Finite Field in a of size 3^4
        sage: a.is_one()
        False
        sage: k(1).is_one()
        True

  `is_square`()

    Return `True` if `self` is a square in `self.parent()`

    ALGORITHM:

    Elements are stored as powers of generators, so we simply check to see if it is an even power of a generator.

    EXAMPLES:

        sage: k.<a> = GF(9); k
        Finite Field in a of size 3^2
        sage: a.is_square()
        False
        sage: v = set([x^2 for x in k])
        sage: [x.is_square() for x in v]
        [True, True, True, True, True]
        sage: [x.is_square() for x in k if not x in v]
        [False, False, False, False]

    TESTS:

        sage: K = GF(27, 'a')
        sage: set([a*a for a in K]) == set([a for a in K if a.is_square()])
        True
        sage: K = GF(25, 'a')
        sage: set([a*a for a in K]) == set([a for a in K if a.is_square()])
        True
        sage: K = GF(16, 'a')
        sage: set([a*a for a in K]) == set([a for a in K if a.is_square()])
        True

  `is_unit`()

    Return `True` if self is nonzero, so it is a unit as an element of the finite field.

    EXAMPLES:

        sage: k.<a> = GF(3^4); k
        Finite Field in a of size 3^4
        sage: a.is_unit()
        True
        sage: k(0).is_unit()
        False

  `log`(base)

    Return the log to the base \(b\) of `self`, i.e., an integer \(n\) such that \(b^n =\) `self`.

    Warning

    TODO – This is currently implemented by solving the discrete log problem – which shouldn’t be needed because of how finite field elements are represented.

    EXAMPLES:

        sage: k.<b> = GF(5^2); k
        Finite Field in b of size 5^2
        sage: a = b^7
        sage: a.log(b)
        7

  `log_repr`()

    Return the log representation of `self` as a string. See the documentation of the `_element_log_repr` function of the parent field.

    EXAMPLES:

        sage: k.<b> = GF(5^2); k
        Finite Field in b of size 5^2
        sage: (b+2).log_repr()
        '15'

  `log_to_int`()

    Returns the int representation of `self`, as a Sage integer. Use `int(self)` to directly get a Python int.

    Elements of this field are represented as ints in as follows: for \(e \in \GF{p}[x]\) with \(e = a_0 + a_1x + a_2x^2 + \cdots\), \(e\) is represented as: \(n = a_0 + a_1 p + a_2 p^2 + \cdots\).

    EXAMPLES:

        sage: k.<b> = GF(5^2); k
        Finite Field in b of size 5^2
        sage: k(4).log_to_int()
        4
        sage: b.log_to_int()
        5
        sage: type(b.log_to_int())
        <type 'sage.rings.integer.Integer'>

  `multiplicative_order`()

    Return the multiplicative order of this field element.

    EXAMPLES:

        sage: S.<b> = GF(5^2); S
        Finite Field in b of size 5^2
        sage: b.multiplicative_order()
        24
        sage: (b^6).multiplicative_order()
        4

  `poly_repr`()

    Return representation of this finite field element as a polynomial in the generator.

    EXAMPLES:

        sage: k.<b> = GF(5^2); k
        Finite Field in b of size 5^2
        sage: (b+2).poly_repr()
        'b + 2'

  `polynomial`(name=None)

    Return self viewed as a polynomial over `self.parent().prime_subfield()`.

    EXAMPLES:

        sage: k.<b> = GF(5^2); k
        Finite Field in b of size 5^2
        sage: f = (b^2+1).polynomial(); f
        b + 4
        sage: type(f)
        <type 'sage.rings.polynomial.polynomial_zmod_flint.Polynomial_zmod_flint'>
        sage: parent(f)
        Univariate Polynomial Ring in b over Finite Field of size 5

  `sqrt`(extend=False, all=False)

    Return a square root of this finite field element in its parent, if there is one. Otherwise, raise a `ValueError`.

    INPUT:

    - `extend` – bool (default: `True`); if `True`, return a square root in an extension ring, if necessary. Otherwise, raise a `ValueError` if the root is not in the base ring.

      Warning

      this option is not implemented!

    - `all` – bool (default: `False`); if `True`, return all square roots of `self`, instead of just one.

    Warning

    The `extend` option is not implemented (yet).

    ALGORITHM:

    `self` is stored as \(a^k\) for some generator \(a\). Return \(a^{k/2}\) for even \(k\).

    EXAMPLES:

        sage: k.<a> = GF(7^2)
        sage: k(2).sqrt()
        3
        sage: k(3).sqrt()
        2*a + 6
        sage: k(3).sqrt()**2
        3
        sage: k(4).sqrt()
        2
        sage: k.<a> = GF(7^3)
        sage: k(3).sqrt()
        Traceback (most recent call last):
        ...
        ValueError: must be a perfect square.

    TESTS:

        sage: K = GF(49, 'a')
        sage: all([a.sqrt()*a.sqrt() == a for a in K if a.is_square()])
        True
        sage: K = GF(27, 'a')
        sage: all([a.sqrt()*a.sqrt() == a for a in K if a.is_square()])
        True
        sage: K = GF(8, 'a')
        sage: all([a.sqrt()*a.sqrt() == a for a in K if a.is_square()])
        True
        sage: K.<a>=FiniteField(9)
        sage: a.sqrt(extend = False, all = True)
        []

class `sage.rings.finite_rings.element_givaro.``FiniteField_givaro_iterator`

  Bases: `object`

  Iterator over `FiniteField_givaro` elements. We iterate multiplicatively, as powers of a fixed internal generator.

  EXAMPLES:

      sage: for x in GF(2^2,'a'): print x
      0
      a
      a + 1
      1

  `next`()

    x.next() -> the next value, or raise StopIteration

`sage.rings.finite_rings.element_givaro.``unpickle_Cache_givaro`(parent, p, k, modulus, rep, cache)

  EXAMPLES:

      sage: k = GF(3**7, 'a')
      sage: loads(dumps(k)) == k # indirect doctest
      True

`sage.rings.finite_rings.element_givaro.``unpickle_FiniteField_givaroElement`(parent, x)

  TESTS:

      sage: k = GF(3**4, 'a')
      sage: e = k.random_element()
      sage: TestSuite(e).run() # indirect doctest

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