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 vastly faster than the PARI implementation which uses polynomials to represent finite field elements. Some functionality in this class however is implemented using the PARI implementation.

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

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 – you may provide a polynomial to use for reduction or one of the following strings:

• 'conway' – force the use of a Conway polynomial, will raise a RuntimeError if none is found in the database
• 'random' – use a random irreducible polynomial
• 'default' – a Conway polynomial is used if found. Otherwise a random polynomial is used

Furthermore, for binary fields we allow two more options:

• 'minimal_weight' – use a minimal weight polynomial, should result in faster arithmetic;
• 'first_lexicographic' – use the first irreducible polynomial in lexicographic order.
• repr – (default: ‘poly’) controls the way elements are printed to the user:

• 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


Three different representations are possible:

sage: sage.rings.finite_rings.finite_field_givaro.FiniteField_givaro(9,repr='poly').gen()
a
sage: sage.rings.finite_rings.finite_field_givaro.FiniteField_givaro(9,repr='int').gen()
3
sage: sage.rings.finite_rings.finite_field_givaro.FiniteField_givaro(9,repr='log').gen()
1

a_times_b_minus_c(a, b, c)

Return a*b - c.

INPUT:

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:

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:

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(2**8, 'a')
sage: e = k.vector_space().gen(1); e
(0, 1, 0, 0, 0, 0, 0, 0)
sage: k(e) #indirect doctest
a


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

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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