예제 #1
0
    def __init__(self, q, name, modulus=None):
        """
        Create finite field of order `q` with variable printed as name.
            
        EXAMPLES::

            sage: from sage.rings.finite_rings.finite_field_ext_pari import FiniteField_ext_pari
            sage: k = FiniteField_ext_pari(9, 'a'); k
            Finite Field in a of size 3^2
        """
        if element_ext_pari.dynamic_FiniteField_ext_pariElement is None: element_ext_pari._late_import()
        from constructor import FiniteField as GF
        q = integer.Integer(q)
        if q < 2:
            raise ArithmeticError, "q must be a prime power"
        from sage.structure.proof.all import arithmetic
        proof = arithmetic()
        if proof:
            F = q.factor()
        else:
            from sage.rings.arith import is_pseudoprime_small_power
            F = is_pseudoprime_small_power(q, get_data=True)
        if len(F) != 1:
            raise ArithmeticError, "q must be a prime power"

        if F[0][1] > 1:
            base_ring = GF(F[0][0])
        else:
            raise ValueError, "The size of the finite field must not be prime."
            #base_ring = self
            
        FiniteField_generic.__init__(self, base_ring, name, normalize=True)

        self._kwargs = {}
        self.__char = F[0][0]
        self.__pari_one = pari.pari(1).Mod(self.__char)
        self.__degree = integer.Integer(F[0][1])
        self.__order = q
        self.__is_field = True

        if modulus is None or modulus == "default":
            from constructor import exists_conway_polynomial
            if exists_conway_polynomial(self.__char, self.__degree):
                modulus = "conway"
            else:
                modulus = "random"

        if isinstance(modulus,str):
            if modulus == "conway":
                from constructor import conway_polynomial
                modulus = conway_polynomial(self.__char, self.__degree)
            elif modulus == "random":
                # The following is fast/deterministic, but has serious problems since
                # it crashes on 64-bit machines, and I can't figure out why:
                #     self.__pari_modulus = pari.pari.finitefield_init(self.__char, self.__degree, self.variable_name())
                # So instead we iterate through random polys until we find an irreducible one.

                R = GF(self.__char)['x']
                while True:
                    modulus = R.random_element(self.__degree)
                    modulus = modulus.monic()
                    if modulus.degree() == self.__degree and modulus.is_irreducible():
                        break
            else:
                raise ValueError("Modulus parameter not understood")

        elif isinstance(modulus, (list, tuple)):
            modulus = GF(self.__char)['x'](modulus)
        elif sage.rings.polynomial.polynomial_element.is_Polynomial(modulus):
            if modulus.parent() is not base_ring:
                modulus = modulus.change_ring(base_ring)
        else:
            raise ValueError("Modulus parameter not understood")

        self.__modulus = modulus
        f = pari.pari(str(modulus))
        self.__pari_modulus = f.subst(modulus.parent().variable_name(), 'a') * self.__pari_one
        self.__gen = element_ext_pari.FiniteField_ext_pariElement(self, pari.pari('a'))

        self._zero_element = self._element_constructor_(0)
        self._one_element = self._element_constructor_(1)
예제 #2
0
    def __init__(self, q, name, modulus=None):
        """
        Create finite field of order q with variable printed as name.

        INPUT:

        - ``q`` -- integer, size of the finite field, not prime
        - ``name`` -- variable used for printing element of the finite
                    field.  Also, two finite fields are considered
                    equal if they have the same variable name, and not
                    otherwise.
        - ``modulus`` -- you may provide a polynomial to use for reduction or
                     a string:
                     '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.
                    
        OUTPUT:

        - FiniteField_ext_pari -- a finite field of order q with given variable name.
            
        EXAMPLES::

            sage: FiniteField(65537)
            Finite Field of size 65537
            sage: FiniteField(2^20, 'c')
            Finite Field in c of size 2^20
            sage: FiniteField(3^11, "b")
            Finite Field in b of size 3^11
            sage: FiniteField(3^11, "b").gen()
            b

        You can also create a finite field using GF, which is a synonym 
        for FiniteField. ::

            sage: GF(19^5, 'a')
            Finite Field in a of size 19^5
        """
        if element_ext_pari.dynamic_FiniteField_ext_pariElement is None:
            element_ext_pari._late_import()
        from constructor import FiniteField as GF
        q = integer.Integer(q)
        if q < 2:
            raise ArithmeticError, "q must be a prime power"
        from sage.structure.proof.all import arithmetic
        proof = arithmetic()
        if proof:
            F = q.factor()
        else:
            from sage.rings.arith import is_pseudoprime_small_power
            F = is_pseudoprime_small_power(q, get_data=True)
        if len(F) != 1:
            raise ArithmeticError, "q must be a prime power"

        if F[0][1] > 1:
            base_ring = GF(F[0][0])
        else:
            raise ValueError, "The size of the finite field must not be prime."
            #base_ring = self

        FiniteField_generic.__init__(self, base_ring, name, normalize=True)

        self._kwargs = {}
        self.__char = F[0][0]
        self.__pari_one = pari.pari(1).Mod(self.__char)
        self.__degree = integer.Integer(F[0][1])
        self.__order = q
        self.__is_field = True

        if modulus is None or modulus == "default":
            from constructor import exists_conway_polynomial
            if exists_conway_polynomial(self.__char, self.__degree):
                modulus = "conway"
            else:
                modulus = "random"

        if isinstance(modulus, str):
            if modulus == "conway":
                from constructor import conway_polynomial
                modulus = conway_polynomial(self.__char, self.__degree)
            elif modulus == "random":
                # The following is fast/deterministic, but has serious problems since
                # it crashes on 64-bit machines, and I can't figure out why:
                #     self.__pari_modulus = pari.pari.finitefield_init(self.__char, self.__degree, self.variable_name())
                # So instead we iterate through random polys until we find an irreducible one.

                R = GF(self.__char)['x']
                while True:
                    modulus = R.random_element(self.__degree)
                    modulus = modulus.monic()
                    if modulus.degree(
                    ) == self.__degree and modulus.is_irreducible():
                        break
            else:
                raise ValueError("Modulus parameter not understood")

        elif isinstance(modulus, (list, tuple)):
            modulus = GF(self.__char)['x'](modulus)
        elif sage.rings.polynomial.polynomial_element.is_Polynomial(modulus):
            if modulus.parent() is not base_ring:
                modulus = modulus.change_ring(base_ring)
        else:
            raise ValueError("Modulus parameter not understood")

        self.__modulus = modulus
        f = pari.pari(str(modulus))
        self.__pari_modulus = f.subst(modulus.parent().variable_name(),
                                      'a') * self.__pari_one
        self.__gen = element_ext_pari.FiniteField_ext_pariElement(
            self, pari.pari('a'))

        self._zero_element = self._element_constructor_(0)
        self._one_element = self._element_constructor_(1)
예제 #3
0
    def __init__(self, q, name, modulus=None):
        """
        Create finite field of order q with variable printed as name.

        INPUT:

        - ``q`` -- integer, size of the finite field, not prime
        - ``name`` -- variable used for printing element of the finite
                    field.  Also, two finite fields are considered
                    equal if they have the same variable name, and not
                    otherwise.
        - ``modulus`` -- you may provide a polynomial to use for reduction or
                     a string:
                     '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.
                    
        OUTPUT:

        - FiniteField_ext_pari -- a finite field of order q with given variable name.
            
        EXAMPLES::

            sage: FiniteField(65537)
            Finite Field of size 65537
            sage: FiniteField(2^20, 'c')
            Finite Field in c of size 2^20
            sage: FiniteField(3^11, "b")
            Finite Field in b of size 3^11
            sage: FiniteField(3^11, "b").gen()
            b

        You can also create a finite field using GF, which is a synonym 
        for FiniteField. ::

            sage: GF(19^5, 'a')
            Finite Field in a of size 19^5
        """
        if element_ext_pari.dynamic_FiniteField_ext_pariElement is None: element_ext_pari._late_import()
        from constructor import FiniteField as GF
        q = integer.Integer(q)
        if q < 2:
            raise ArithmeticError, "q must be a prime power"
        from sage.structure.proof.all import arithmetic
        proof = arithmetic()
        if proof:
            F = q.factor()
        else:
            from sage.rings.arith import is_pseudoprime_small_power
            F = is_pseudoprime_small_power(q, get_data=True)
        if len(F) != 1:
            raise ArithmeticError, "q must be a prime power"

        if F[0][1] > 1:
            base_ring = GF(F[0][0])
        else:
            raise ValueError, "The size of the finite field must not be prime."
            #base_ring = self
            
        FiniteField_generic.__init__(self, base_ring, name, normalize=True)

        self._kwargs = {}
        self.__char = F[0][0]
        self.__pari_one = pari.pari(1).Mod(self.__char)
        self.__degree = integer.Integer(F[0][1])
        self.__order = q
        self.__is_field = True

        if modulus is None or modulus == "default":
            from constructor import exists_conway_polynomial
            if exists_conway_polynomial(self.__char, self.__degree):
                modulus = "conway"
            else:
                modulus = "random"

        if isinstance(modulus,str):
            if modulus == "conway":
                from constructor import conway_polynomial
                modulus = conway_polynomial(self.__char, self.__degree)
            elif modulus == "random":
                # The following is fast/deterministic, but has serious problems since
                # it crashes on 64-bit machines, and I can't figure out why:
                #     self.__pari_modulus = pari.pari.finitefield_init(self.__char, self.__degree, self.variable_name())
                # So instead we iterate through random polys until we find an irreducible one.

                R = GF(self.__char)['x']
                while True:
                    modulus = R.random_element(self.__degree)
                    modulus = modulus.monic()
                    if modulus.degree() == self.__degree and modulus.is_irreducible():
                        break
            else:
                raise ValueError("Modulus parameter not understood")

        elif isinstance(modulus, (list, tuple)):
            modulus = GF(self.__char)['x'](modulus)
        elif sage.rings.polynomial.polynomial_element.is_Polynomial(modulus):
            if modulus.parent() is not base_ring:
                modulus = modulus.change_ring(base_ring)
        else:
            raise ValueError("Modulus parameter not understood")

        self.__modulus = modulus
        f = pari.pari(str(modulus))
        self.__pari_modulus = f.subst(modulus.parent().variable_name(), 'a') * self.__pari_one
        self.__gen = element_ext_pari.FiniteField_ext_pariElement(self, pari.pari('a'))

        self._zero_element = self._element_constructor_(0)
        self._one_element = self._element_constructor_(1)
예제 #4
0
    def __init__(self, q, name, modulus=None):
        """
        Create finite field of order `q` with variable printed as name.

        EXAMPLES::

            sage: from sage.rings.finite_rings.finite_field_ext_pari import FiniteField_ext_pari
            sage: k = FiniteField_ext_pari(9, 'a'); k
            Finite Field in a of size 3^2
        """
        if element_ext_pari.dynamic_FiniteField_ext_pariElement is None:
            element_ext_pari._late_import()
        from constructor import FiniteField as GF
        q = integer.Integer(q)
        if q < 2:
            raise ArithmeticError, "q must be a prime power"
        from sage.structure.proof.all import arithmetic
        proof = arithmetic()
        if proof:
            F = q.factor()
        else:
            from sage.rings.arith import is_pseudoprime_small_power
            F = is_pseudoprime_small_power(q, get_data=True)
        if len(F) != 1:
            raise ArithmeticError, "q must be a prime power"

        if F[0][1] > 1:
            base_ring = GF(F[0][0])
        else:
            raise ValueError, "The size of the finite field must not be prime."
            #base_ring = self

        FiniteField_generic.__init__(self, base_ring, name, normalize=True)

        self._kwargs = {}
        self.__char = F[0][0]
        self.__pari_one = pari.pari(1).Mod(self.__char)
        self.__degree = integer.Integer(F[0][1])
        self.__order = q
        self.__is_field = True

        if modulus is None or modulus == "default":
            from constructor import exists_conway_polynomial
            if exists_conway_polynomial(self.__char, self.__degree):
                modulus = "conway"
            else:
                modulus = "random"

        if isinstance(modulus, str):
            if modulus == "conway":
                from constructor import conway_polynomial
                modulus = conway_polynomial(self.__char, self.__degree)
            elif modulus == "random":
                # The following is fast/deterministic, but has serious problems since
                # it crashes on 64-bit machines, and I can't figure out why:
                #     self.__pari_modulus = pari.pari.finitefield_init(self.__char, self.__degree, self.variable_name())
                # So instead we iterate through random polys until we find an irreducible one.

                R = GF(self.__char)['x']
                while True:
                    modulus = R.random_element(self.__degree)
                    modulus = modulus.monic()
                    if modulus.degree(
                    ) == self.__degree and modulus.is_irreducible():
                        break
            else:
                raise ValueError("Modulus parameter not understood")

        elif isinstance(modulus, (list, tuple)):
            modulus = GF(self.__char)['x'](modulus)
        elif sage.rings.polynomial.polynomial_element.is_Polynomial(modulus):
            if modulus.parent() is not base_ring:
                modulus = modulus.change_ring(base_ring)
        else:
            raise ValueError("Modulus parameter not understood")

        self.__modulus = modulus
        f = pari.pari(str(modulus))
        self.__pari_modulus = f.subst(modulus.parent().variable_name(),
                                      'a') * self.__pari_one
        self.__gen = element_ext_pari.FiniteField_ext_pariElement(
            self, pari.pari('a'))

        self._zero_element = self._element_constructor_(0)
        self._one_element = self._element_constructor_(1)