def __init__(self, q, name="a", modulus=None, repr="poly", cache=False): """ Initialize ``self``. EXAMPLES:: sage: k.<a> = GF(2^3) sage: j.<b> = GF(3^4) sage: k == j False sage: GF(2^3,'a') == copy(GF(2^3,'a')) True sage: TestSuite(GF(2^3, 'a')).run() """ self._kwargs = {} if repr not in ['int', 'log', 'poly']: raise ValueError("Unknown representation %s" % repr) q = Integer(q) if q < 2: raise ValueError("q must be a prime power") F = q.factor() if len(F) > 1: raise ValueError("q must be a prime power") p = F[0][0] k = F[0][1] if q >= 1 << 16: raise ValueError("q must be < 2^16") from constructor import GF FiniteField.__init__(self, GF(p), name, normalize=False) self._kwargs['repr'] = repr self._kwargs['cache'] = cache from sage.rings.polynomial.polynomial_element import is_Polynomial if not is_Polynomial(modulus): from sage.misc.superseded import deprecation deprecation( 16930, "constructing a FiniteField_givaro without giving a polynomial as modulus is deprecated, use the more general FiniteField constructor instead" ) R = GF(p)['x'] if modulus is None or isinstance(modulus, str): modulus = R.irreducible_element(k, algorithm=modulus) else: modulus = R(modulus) self._cache = Cache_givaro(self, p, k, modulus, repr, cache) self._modulus = modulus
def __init__(self, q, name="a", modulus=None, repr="poly", cache=False): """ Initialize ``self``. EXAMPLES:: sage: k.<a> = GF(2^3) sage: j.<b> = GF(3^4) sage: k == j False sage: GF(2^3,'a') == copy(GF(2^3,'a')) True sage: TestSuite(GF(2^3, 'a')).run() """ self._kwargs = {} if repr not in ['int', 'log', 'poly']: raise ValueError("Unknown representation %s"%repr) q = Integer(q) if q < 2: raise ValueError("q must be a prime power") F = q.factor() if len(F) > 1: raise ValueError("q must be a prime power") p = F[0][0] k = F[0][1] if q >= 1<<16: raise ValueError("q must be < 2^16") from constructor import GF FiniteField.__init__(self, GF(p), name, normalize=False) self._kwargs['repr'] = repr self._kwargs['cache'] = cache from sage.rings.polynomial.polynomial_element import is_Polynomial if not is_Polynomial(modulus): from sage.misc.superseded import deprecation deprecation(16930, "constructing a FiniteField_givaro without giving a polynomial as modulus is deprecated, use the more general FiniteField constructor instead") R = GF(p)['x'] if modulus is None or isinstance(modulus, str): modulus = R.irreducible_element(k, algorithm=modulus) else: modulus = R(modulus) self._cache = Cache_givaro(self, p, k, modulus, repr, cache) self._modulus = modulus
def __init__(self, p, modulus, name=None): """ Create a finite field of characteristic `p` defined by the polynomial ``modulus``, with distinguished generator called ``name``. EXAMPLE:: sage: from sage.rings.finite_rings.finite_field_pari_ffelt import FiniteField_pari_ffelt sage: R.<x> = PolynomialRing(GF(3)) sage: k = FiniteField_pari_ffelt(3, x^2 + 2*x + 2, 'a'); k Finite Field in a of size 3^2 """ n = modulus.degree() if n < 2: raise ValueError("the degree must be at least 2") FiniteField.__init__(self, base=GF(p), names=name, normalize=True) self._modulus = modulus self._degree = n self._kwargs = {} self._gen_pari = modulus._pari_with_name(self._names[0]).ffgen() self._zero_element = self.element_class(self, 0) self._one_element = self.element_class(self, 1) self._gen = self.element_class(self, self._gen_pari)
def polynomial(self, name=None): """ Return the irreducible characteristic polynomial of the generator of this finite field, i.e., the polynomial `f(x)` so elements of the finite field as elements modulo `f`. EXAMPLES:: sage: k = FiniteField(17) sage: k.polynomial('x') x sage: from sage.rings.finite_rings.finite_field_ext_pari import FiniteField_ext_pari sage: k = FiniteField_ext_pari(9,'a') sage: k.polynomial('x') x^2 + 2*x + 2 """ if name is None: name = self.variable_name() try: return self.__polynomial[name] except (AttributeError, KeyError): from constructor import FiniteField as GF R = GF(self.characteristic())[name] f = R(self._pari_modulus()) try: self.__polynomial[name] = f except (KeyError, AttributeError): self.__polynomial = {} self.__polynomial[name] = f return f
def prime_subfield(self): r""" 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'> """ try: return self._prime_subfield except AttributeError: from constructor import GF self._prime_subfield = GF(self.characteristic()) return self._prime_subfield
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 conway_polynomials 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 conway_polynomials 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)
def __init__(self, q, name, modulus=None): """ Create finite field of order `q` with variable printed as name. EXAMPLES:: sage: k = FiniteField(9, 'a', impl='pari_mod'); k Finite Field in a of size 3^2 """ from sage.misc.superseded import deprecation deprecation(17297, 'The "pari_mod" finite field implementation is deprecated') 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") # note: the following call takes care of the fact that # proof.arithmetic() is True or False. p, n = q.is_prime_power(get_data=True) if n > 1: base_ring = GF(p) elif n == 0: raise ArithmeticError("q must be a prime power") else: raise ValueError("The size of the finite field must not be prime.") FiniteField_generic.__init__(self, base_ring, name, normalize=True) self._kwargs = {} self.__char = p self.__pari_one = pari.pari(1).Mod(self.__char) self.__degree = n self.__order = q self.__is_field = True if not sage.rings.polynomial.polynomial_element.is_Polynomial(modulus): from sage.misc.superseded import deprecation deprecation(16930, "constructing a FiniteField_ext_pari without giving a polynomial as modulus is deprecated, use the more general FiniteField constructor instead") if modulus is None or modulus == "default": from conway_polynomials 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 conway_polynomials 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.base_ring() 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)
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)