def create_key_and_extra_args(self, order, name=None, modulus=None, names=None, impl=None, proof=None, **kwds): """ EXAMPLES:: sage: GF.create_key_and_extra_args(9, 'a') ((9, ('a',), 'conway', None, '{}', 3, 2, True), {}) sage: GF.create_key_and_extra_args(9, 'a', foo='value') ((9, ('a',), 'conway', None, "{'foo': 'value'}", 3, 2, True), {'foo': 'value'}) """ from sage.structure.proof.all import WithProof, arithmetic if proof is None: proof = arithmetic() with WithProof('arithmetic', proof): order = int(order) if order <= 1: raise ValueError("the order of a finite field must be > 1.") if arith.is_prime(order): name = None modulus = None p = integer.Integer(order) n = integer.Integer(1) elif arith.is_prime_power(order): if not names is None: name = names name = normalize_names(1, name) p, n = arith.factor(order)[0] if modulus is None or modulus == "default": if exists_conway_polynomial(p, n): modulus = "conway" else: if p == 2: modulus = "minimal_weight" else: modulus = "random" elif modulus == "random": modulus += str(random.randint(0, 1 << 128)) if isinstance(modulus, (list, tuple)): modulus = FiniteField(p)['x'](modulus) # some classes use 'random' as the modulus to # generate a random modulus, but we don't want # to cache it elif sage.rings.polynomial.polynomial_element.is_Polynomial( modulus): modulus = modulus.change_variable_name('x') elif not isinstance(modulus, str): raise ValueError("Modulus parameter not understood.") else: # Neither a prime, nor a prime power raise ValueError( "the order of a finite field must be a prime power.") return (order, name, modulus, impl, str(kwds), p, n, proof), kwds
def create_key_and_extra_args(self, order, name=None, modulus=None, names=None, impl=None, proof=None, **kwds): """ EXAMPLES:: sage: GF.create_key_and_extra_args(9, 'a') ((9, ('a',), x^2 + 2*x + 2, None, '{}', 3, 2, True), {}) sage: GF.create_key_and_extra_args(9, 'a', foo='value') ((9, ('a',), x^2 + 2*x + 2, None, "{'foo': 'value'}", 3, 2, True), {'foo': 'value'}) """ from sage.structure.proof.all import WithProof, arithmetic if proof is None: proof = arithmetic() with WithProof('arithmetic', proof): order = int(order) if order <= 1: raise ValueError("the order of a finite field must be > 1.") if arith.is_prime(order): name = None modulus = None p = integer.Integer(order) n = integer.Integer(1) elif arith.is_prime_power(order): if not names is None: name = names name = normalize_names(1, name) p, n = arith.factor(order)[0] if modulus is None or isinstance(modulus, str): # A string specifies an algorithm to find a suitable modulus. if modulus == "default": # for backward compatibility modulus = None modulus = GF(p)['x'].irreducible_element(n, algorithm=modulus) elif isinstance(modulus, (list, tuple)): modulus = GF(p)['x'](modulus) elif sage.rings.polynomial.polynomial_element.is_Polynomial( modulus): modulus = modulus.change_variable_name('x') else: raise TypeError("wrong type for modulus parameter") else: raise ValueError( "the order of a finite field must be a prime power.") return (order, name, modulus, impl, str(kwds), p, n, proof), kwds
def other_keys(self, key, K): """ EXAMPLES:: sage: key, extra = GF.create_key_and_extra_args(9, 'a'); key (9, ('a',), x^2 + 2*x + 2, None, '{}', 3, 2, True) sage: K = GF.create_object(0, key); K Finite Field in a of size 3^2 sage: GF.other_keys(key, K) [(9, ('a',), x^2 + 2*x + 2, None, '{}', 3, 2, True), (9, ('a',), x^2 + 2*x + 2, 'givaro', '{}', 3, 2, True)] """ if len(key) == 5: # backward compat order, name, modulus, impl, _ = key p, n = arith.factor(order)[0] proof = True else: order, name, modulus, impl, _, p, n, proof = key from sage.structure.proof.all import WithProof with WithProof('arithmetic', proof): if K.degree() > 1: modulus = K.modulus().change_variable_name('x') new_keys = [(order, name, modulus, impl, _, p, n, proof)] from finite_field_prime_modn import FiniteField_prime_modn if isinstance(K, FiniteField_prime_modn): impl = 'modn' elif isinstance(K, FiniteField_givaro): impl = 'givaro' else: from finite_field_ntl_gf2e import FiniteField_ntl_gf2e from finite_field_ext_pari import FiniteField_ext_pari from finite_field_pari_ffelt import FiniteField_pari_ffelt if isinstance(K, FiniteField_ntl_gf2e): impl = 'ntl' elif isinstance(K, FiniteField_ext_pari): impl = 'pari_mod' elif isinstance(K, FiniteField_pari_ffelt): impl = 'pari_ffelt' new_keys.append((order, name, modulus, impl, _, p, n, proof)) return new_keys
def create_object(self, version, key, **kwds): """ EXAMPLES:: sage: K = GF(19) # indirect doctest sage: TestSuite(K).run() We try to create finite fields with various implementations:: sage: k = GF(2, impl='modn') sage: k = GF(2, impl='givaro') sage: k = GF(2, impl='ntl') sage: k = GF(2, impl='pari') Traceback (most recent call last): ... ValueError: the degree must be at least 2 sage: k = GF(2, impl='supercalifragilisticexpialidocious') Traceback (most recent call last): ... ValueError: no such finite field implementation: 'supercalifragilisticexpialidocious' sage: k.<a> = GF(2^15, impl='modn') Traceback (most recent call last): ... ValueError: the 'modn' implementation requires a prime order sage: k.<a> = GF(2^15, impl='givaro') sage: k.<a> = GF(2^15, impl='ntl') sage: k.<a> = GF(2^15, impl='pari') sage: k.<a> = GF(3^60, impl='modn') Traceback (most recent call last): ... ValueError: the 'modn' implementation requires a prime order sage: k.<a> = GF(3^60, impl='givaro') Traceback (most recent call last): ... ValueError: q must be < 2^16 sage: k.<a> = GF(3^60, impl='ntl') Traceback (most recent call last): ... ValueError: q must be a 2-power sage: k.<a> = GF(3^60, impl='pari') """ # IMPORTANT! If you add a new class to the list of classes # that get cached by this factor object, then you *must* add # the following method to that class in order to fully support # pickling: # # def __reduce__(self): # and include good doctests, please! # return self._factory_data[0].reduce_data(self) # # This is not in the base class for finite fields, since some finite # fields need not be created using this factory object, e.g., residue # class fields. if len(key) == 5: # for backward compatibility of pickles (see trac 10975). order, name, modulus, impl, _ = key p, n = Integer(order).factor()[0] proof = True prefix = kwds.get('prefix', None) # We can set the defaults here to be those for givaro # as they are otherwise ignored repr = 'poly' elem_cache = (order < 500) elif len(key) == 8: # For backward compatibility of pickles (see trac #21433) order, name, modulus, impl, _, p, n, proof = key prefix = kwds.get('prefix', None) # We can set the defaults here to be those for givaro # as they are otherwise ignored repr = kwds.get('repr', 'poly') elem_cache = kwds.get('elem_cache', (order < 500)) else: order, name, modulus, impl, p, n, proof, prefix, repr, elem_cache = key if impl == 'modn': if n != 1: raise ValueError( "the 'modn' implementation requires a prime order") from .finite_field_prime_modn import FiniteField_prime_modn # Using a check option here is probably a worthwhile # compromise since this constructor is simple and used a # huge amount. K = FiniteField_prime_modn(order, check=False, modulus=modulus) else: # We have to do this with block so that the finite field # constructors below will use the proof flag that was # passed in when checking for primality, factoring, etc. # Otherwise, we would have to complicate all of their # constructors with check options. from sage.structure.proof.all import WithProof with WithProof('arithmetic', proof): if impl == 'givaro': K = FiniteField_givaro(order, name, modulus, repr, elem_cache) elif impl == 'ntl': from .finite_field_ntl_gf2e import FiniteField_ntl_gf2e K = FiniteField_ntl_gf2e(order, name, modulus) elif impl == 'pari_ffelt' or impl == 'pari': from .finite_field_pari_ffelt import FiniteField_pari_ffelt K = FiniteField_pari_ffelt(p, modulus, name) else: raise ValueError( "no such finite field implementation: %r" % impl) # Temporary; see create_key_and_extra_args() above. if prefix is not None: K._prefix = prefix return K
def create_key_and_extra_args(self, order, name=None, modulus=None, names=None, impl=None, proof=None, check_irreducible=True, prefix=None, repr=None, elem_cache=None, **kwds): """ EXAMPLES:: sage: GF.create_key_and_extra_args(9, 'a') ((9, ('a',), x^2 + 2*x + 2, 'givaro', 3, 2, True, None, 'poly', True), {}) We do not take invalid keyword arguments and raise a value error to better ensure uniqueness:: sage: GF.create_key_and_extra_args(9, 'a', foo='value') Traceback (most recent call last): ... TypeError: create_key_and_extra_args() got an unexpected keyword argument 'foo' Moreover, ``repr`` and ``elem_cache`` are ignored when not using givaro:: sage: GF.create_key_and_extra_args(16, 'a', impl='ntl', repr='poly') ((16, ('a',), x^4 + x + 1, 'ntl', 2, 4, True, None, None, None), {}) sage: GF.create_key_and_extra_args(16, 'a', impl='ntl', elem_cache=False) ((16, ('a',), x^4 + x + 1, 'ntl', 2, 4, True, None, None, None), {}) sage: GF(16, impl='ntl') is GF(16, impl='ntl', repr='foo') True We handle extra arguments for the givaro finite field and create unique objects for their defaults:: sage: GF(25, impl='givaro') is GF(25, impl='givaro', repr='poly') True sage: GF(25, impl='givaro') is GF(25, impl='givaro', elem_cache=True) True sage: GF(625, impl='givaro') is GF(625, impl='givaro', elem_cache=False) True We explicitly take ``structure``, ``implementation`` and ``prec`` attributes for compatibility with :class:`~sage.categories.pushout.AlgebraicExtensionFunctor` but we ignore them as they are not used, see :trac:`21433`:: sage: GF.create_key_and_extra_args(9, 'a', structure=None) ((9, ('a',), x^2 + 2*x + 2, 'givaro', 3, 2, True, None, 'poly', True), {}) """ import sage.arith.all from sage.structure.proof.all import WithProof, arithmetic if proof is None: proof = arithmetic() for key, val in kwds.items(): if key not in ['structure', 'implementation', 'prec', 'embedding']: raise TypeError( "create_key_and_extra_args() got an unexpected keyword argument '%s'" % key) if not (val is None or isinstance(val, list) and all(c is None for c in val)): raise NotImplementedError( "ring extension with prescribed %s is not implemented" % key) with WithProof('arithmetic', proof): order = Integer(order) if order <= 1: raise ValueError( "the order of a finite field must be at least 2") if order.is_prime(): p = order n = Integer(1) if impl is None: impl = 'modn' name = ('x', ) # Ignore name # Every polynomial of degree 1 is irreducible check_irreducible = False elif order.is_prime_power(): if names is not None: name = names if name is not None: name = normalize_names(1, name) p, n = order.factor()[0] if name is None: if prefix is None: prefix = 'z' name = prefix + str(n) if modulus is not None: raise ValueError( "no modulus may be specified if variable name not given" ) # Fpbar will have a strong reference, since algebraic_closure caches its results, # and the coefficients of modulus lie in GF(p) Fpbar = GF(p).algebraic_closure(prefix) # This will give a Conway polynomial if p,n is small enough to be in the database # and a pseudo-Conway polynomial if it's not. modulus = Fpbar._get_polynomial(n) check_irreducible = False if impl is None: if order < zech_log_bound: impl = 'givaro' elif p == 2: impl = 'ntl' else: impl = 'pari_ffelt' else: raise ValueError( "the order of a finite field must be a prime power") # Determine modulus. # For the 'modn' implementation, we use the following # optimization which we also need to avoid an infinite loop: # a modulus of None is a shorthand for x-1. if modulus is not None or impl != 'modn': R = PolynomialRing(FiniteField(p), 'x') if modulus is None: modulus = R.irreducible_element(n) if isinstance(modulus, str): # A string specifies an algorithm to find a suitable modulus. modulus = R.irreducible_element(n, algorithm=modulus) else: if sage.rings.polynomial.polynomial_element.is_Polynomial( modulus): modulus = modulus.change_variable_name('x') modulus = R(modulus).monic() if modulus.degree() != n: raise ValueError( "the degree of the modulus does not equal the degree of the field" ) if check_irreducible and not modulus.is_irreducible(): raise ValueError( "finite field modulus must be irreducible but it is not" ) # If modulus is x - 1 for impl="modn", set it to None if impl == 'modn' and modulus[0] == -1: modulus = None # Check extra arguments for givaro and setup their defaults # TODO: ntl takes a repr, but ignores it if impl == 'givaro': if repr is None: repr = 'poly' if elem_cache is None: elem_cache = (order < 500) else: # This has the effect of ignoring these keywords repr = None elem_cache = None return (order, name, modulus, impl, p, n, proof, prefix, repr, elem_cache), {}
def create_object(self, version, key, check_irreducible=True, elem_cache=None, names=None, **kwds): """ EXAMPLES:: sage: K = GF(19) sage: TestSuite(K).run() """ # IMPORTANT! If you add a new class to the list of classes # that get cached by this factor object, then you *must* add # the following method to that class in order to fully support # pickling: # # def __reduce__(self): # and include good doctests, please! # return self._factory_data[0].reduce_data(self) # # This is not in the base class for finite fields, since some finite # fields need not be created using this factory object, e.g., residue # class fields. if len(key) == 5: # for backward compatibility of pickles (see trac 10975). order, name, modulus, impl, _ = key p, n = arith.factor(order)[0] proof = True else: order, name, modulus, impl, _, p, n, proof = key if isinstance(modulus, str) and modulus.startswith("random"): modulus = "random" if elem_cache is None: elem_cache = order < 500 if n == 1 and (impl is None or impl == 'modn'): from finite_field_prime_modn import FiniteField_prime_modn # Using a check option here is probably a worthwhile # compromise since this constructor is simple and used a # huge amount. K = FiniteField_prime_modn(order, check=False, **kwds) else: # We have to do this with block so that the finite field # constructors below will use the proof flag that was # passed in when checking for primality, factoring, etc. # Otherwise, we would have to complicate all of their # constructors with check options (like above). from sage.structure.proof.all import WithProof with WithProof('arithmetic', proof): if check_irreducible and polynomial_element.is_Polynomial( modulus): if modulus.parent().base_ring().characteristic() == 0: modulus = modulus.change_ring(FiniteField(p)) if not modulus.is_irreducible(): raise ValueError( "finite field modulus must be irreducible but it is not." ) if modulus.degree() != n: raise ValueError( "The degree of the modulus does not correspond to the cardinality of the field." ) if name is None: raise TypeError("you must specify the generator name.") if order < zech_log_bound: # DO *NOT* use for prime subfield, since that would lead to # a circular reference in the call to ParentWithGens in the # __init__ method. K = FiniteField_givaro(order, name, modulus, cache=elem_cache, **kwds) else: if order % 2 == 0 and (impl is None or impl == 'ntl'): from element_ntl_gf2e import FiniteField_ntl_gf2e K = FiniteField_ntl_gf2e(order, name, modulus, **kwds) else: from finite_field_ext_pari import FiniteField_ext_pari K = FiniteField_ext_pari(order, name, modulus, **kwds) return K
def create_key_and_extra_args(self, order, name=None, modulus=None, names=None, impl=None, proof=None, **kwds): """ EXAMPLES:: sage: GF.create_key_and_extra_args(9, 'a') ((9, ('a',), x^2 + 2*x + 2, None, '{}', 3, 2, True), {}) sage: GF.create_key_and_extra_args(9, 'a', foo='value') ((9, ('a',), x^2 + 2*x + 2, None, "{'foo': 'value'}", 3, 2, True), {'foo': 'value'}) """ from sage.structure.proof.all import WithProof, arithmetic if proof is None: proof = arithmetic() with WithProof('arithmetic', proof): order = int(order) if order <= 1: raise ValueError("the order of a finite field must be > 1.") if arith.is_prime(order): name = None modulus = None p = integer.Integer(order) n = integer.Integer(1) elif arith.is_prime_power(order): if not names is None: name = names name = normalize_names(1, name) p, n = arith.factor(order)[0] # The following is a temporary solution that allows us # to construct compatible systems of finite fields # until algebraic closures of finite fields are # implemented in Sage. It requires the user to # specify two parameters: # # - `conway` -- boolean; if True, this field is # constructed to fit in a compatible system using # a Conway polynomial. # - `prefix` -- a string used to generate names for # automatically constructed finite fields # # See the docstring of FiniteFieldFactory for examples. # # Once algebraic closures of finite fields are # implemented, this syntax should be superseded by # something like the following: # # sage: Fpbar = GF(5).algebraic_closure('z') # sage: F, e = Fpbar.subfield(3) # e is the embedding into Fpbar # sage: F # Finite field in z3 of size 5^3 # # This temporary solution only uses actual Conway # polynomials (no pseudo-Conway polynomials), since # pseudo-Conway polynomials are not unique, and until # we have algebraic closures of finite fields, there # is no good place to store a specific choice of # pseudo-Conway polynomials. if name is None: if not (kwds.has_key('conway') and kwds['conway']): raise ValueError( "parameter 'conway' is required if no name given") if not kwds.has_key('prefix'): raise ValueError( "parameter 'prefix' is required if no name given") name = kwds['prefix'] + str(n) if kwds.has_key('conway') and kwds['conway']: from conway_polynomials import conway_polynomial if not kwds.has_key('prefix'): raise ValueError( "a prefix must be specified if conway=True") if modulus is not None: raise ValueError( "no modulus may be specified if conway=True") # The following raises a RuntimeError if no polynomial is found. modulus = conway_polynomial(p, n) if modulus is None or isinstance(modulus, str): # A string specifies an algorithm to find a suitable modulus. if modulus == "default": # for backward compatibility modulus = None modulus = GF(p)['x'].irreducible_element(n, algorithm=modulus) elif isinstance(modulus, (list, tuple)): modulus = GF(p)['x'](modulus) elif sage.rings.polynomial.polynomial_element.is_Polynomial( modulus): modulus = modulus.change_variable_name('x') else: raise TypeError("wrong type for modulus parameter") else: raise ValueError( "the order of a finite field must be a prime power.") return (order, name, modulus, impl, str(kwds), p, n, proof), kwds
def create_key_and_extra_args(self, order, name=None, modulus=None, names=None, impl=None, proof=None, check_irreducible=True, **kwds): """ EXAMPLES:: sage: GF.create_key_and_extra_args(9, 'a') ((9, ('a',), x^2 + 2*x + 2, 'givaro', '{}', 3, 2, True), {}) sage: GF.create_key_and_extra_args(9, 'a', foo='value') ((9, ('a',), x^2 + 2*x + 2, 'givaro', "{'foo': 'value'}", 3, 2, True), {'foo': 'value'}) """ import sage.arith.all from sage.structure.proof.all import WithProof, arithmetic if proof is None: proof = arithmetic() with WithProof('arithmetic', proof): order = Integer(order) if order <= 1: raise ValueError( "the order of a finite field must be at least 2") if order.is_prime(): p = order n = Integer(1) if impl is None: impl = 'modn' name = ('x', ) # Ignore name # Every polynomial of degree 1 is irreducible check_irreducible = False elif order.is_prime_power(): if names is not None: name = names if name is not None: name = normalize_names(1, name) p, n = order.factor()[0] # The following is a temporary solution that allows us # to construct compatible systems of finite fields # until algebraic closures of finite fields are # implemented in Sage. It requires the user to # specify two parameters: # # - `conway` -- boolean; if True, this field is # constructed to fit in a compatible system using # a Conway polynomial. # - `prefix` -- a string used to generate names for # automatically constructed finite fields # # See the docstring of FiniteFieldFactory for examples. # # Once algebraic closures of finite fields are # implemented, this syntax should be superseded by # something like the following: # # sage: Fpbar = GF(5).algebraic_closure('z') # sage: F, e = Fpbar.subfield(3) # e is the embedding into Fpbar # sage: F # Finite field in z3 of size 5^3 # # This temporary solution only uses actual Conway # polynomials (no pseudo-Conway polynomials), since # pseudo-Conway polynomials are not unique, and until # we have algebraic closures of finite fields, there # is no good place to store a specific choice of # pseudo-Conway polynomials. if name is None: if not ('conway' in kwds and kwds['conway']): raise ValueError( "parameter 'conway' is required if no name given") if 'prefix' not in kwds: raise ValueError( "parameter 'prefix' is required if no name given") name = kwds['prefix'] + str(n) if 'conway' in kwds and kwds['conway']: from conway_polynomials import conway_polynomial if 'prefix' not in kwds: raise ValueError( "a prefix must be specified if conway=True") if modulus is not None: raise ValueError( "no modulus may be specified if conway=True") # The following raises a RuntimeError if no polynomial is found. modulus = conway_polynomial(p, n) if impl is None: if order < zech_log_bound: impl = 'givaro' elif p == 2: impl = 'ntl' else: impl = 'pari_ffelt' else: raise ValueError( "the order of a finite field must be a prime power") # Determine modulus. # For the 'modn' implementation, we use the following # optimization which we also need to avoid an infinite loop: # a modulus of None is a shorthand for x-1. if modulus is not None or impl != 'modn': R = PolynomialRing(FiniteField(p), 'x') if modulus is None: modulus = R.irreducible_element(n) if isinstance(modulus, str): # A string specifies an algorithm to find a suitable modulus. if modulus == "default": from sage.misc.superseded import deprecation deprecation( 16983, "the modulus 'default' is deprecated, use modulus=None instead (which is the default)" ) modulus = None modulus = R.irreducible_element(n, algorithm=modulus) else: if sage.rings.polynomial.polynomial_element.is_Polynomial( modulus): modulus = modulus.change_variable_name('x') modulus = R(modulus).monic() if modulus.degree() != n: raise ValueError( "the degree of the modulus does not equal the degree of the field" ) if check_irreducible and not modulus.is_irreducible(): raise ValueError( "finite field modulus must be irreducible but it is not" ) # If modulus is x - 1 for impl="modn", set it to None if impl == 'modn' and modulus[0] == -1: modulus = None return (order, name, modulus, impl, str(kwds), p, n, proof), kwds
def create_key_and_extra_args(self, order, name=None, modulus=None, names=None, impl=None, proof=None, check_irreducible=True, **kwds): """ EXAMPLES:: sage: GF.create_key_and_extra_args(9, 'a') ((9, ('a',), x^2 + 2*x + 2, 'givaro', '{}', 3, 2, True), {}) sage: GF.create_key_and_extra_args(9, 'a', foo='value') ((9, ('a',), x^2 + 2*x + 2, 'givaro', "{'foo': 'value'}", 3, 2, True), {'foo': 'value'}) """ import sage.arith.all from sage.structure.proof.all import WithProof, arithmetic if proof is None: proof = arithmetic() with WithProof('arithmetic', proof): order = Integer(order) if order <= 1: raise ValueError( "the order of a finite field must be at least 2") if order.is_prime(): p = order n = Integer(1) if impl is None: impl = 'modn' name = ('x', ) # Ignore name # Every polynomial of degree 1 is irreducible check_irreducible = False elif order.is_prime_power(): if names is not None: name = names if name is not None: name = normalize_names(1, name) p, n = order.factor()[0] if name is None: if 'prefix' not in kwds: kwds['prefix'] = 'z' name = kwds['prefix'] + str(n) if modulus is not None: raise ValueError( "no modulus may be specified if variable name not given" ) if 'conway' in kwds: del kwds['conway'] from sage.misc.superseded import deprecation deprecation( 17569, "the 'conway' argument is deprecated, pseudo-conway polynomials are now used by default if no variable name is given" ) # Fpbar will have a strong reference, since algebraic_closure caches its results, # and the coefficients of modulus lie in GF(p) Fpbar = GF(p).algebraic_closure(kwds.get('prefix', 'z')) # This will give a Conway polynomial if p,n is small enough to be in the database # and a pseudo-Conway polynomial if it's not. modulus = Fpbar._get_polynomial(n) check_irreducible = False if impl is None: if order < zech_log_bound: impl = 'givaro' elif p == 2: impl = 'ntl' else: impl = 'pari_ffelt' else: raise ValueError( "the order of a finite field must be a prime power") # Determine modulus. # For the 'modn' implementation, we use the following # optimization which we also need to avoid an infinite loop: # a modulus of None is a shorthand for x-1. if modulus is not None or impl != 'modn': R = PolynomialRing(FiniteField(p), 'x') if modulus is None: modulus = R.irreducible_element(n) if isinstance(modulus, str): # A string specifies an algorithm to find a suitable modulus. if modulus == "default": from sage.misc.superseded import deprecation deprecation( 16983, "the modulus 'default' is deprecated, use modulus=None instead (which is the default)" ) modulus = None modulus = R.irreducible_element(n, algorithm=modulus) else: if sage.rings.polynomial.polynomial_element.is_Polynomial( modulus): modulus = modulus.change_variable_name('x') modulus = R(modulus).monic() if modulus.degree() != n: raise ValueError( "the degree of the modulus does not equal the degree of the field" ) if check_irreducible and not modulus.is_irreducible(): raise ValueError( "finite field modulus must be irreducible but it is not" ) # If modulus is x - 1 for impl="modn", set it to None if impl == 'modn' and modulus[0] == -1: modulus = None return (order, name, modulus, impl, str(kwds), p, n, proof), kwds