def pinch_method(p, n, m, f = None, g = None): ''' Function that given three integers p, n and m such as : - p is prime, - m is dividing n and there exists a primitive mth root that spans F_{p^n} and none of its subfields, returns the image, in a finite field G, of the polynomial generator of a finite field F with the same cardinality as G. ''' c, w = cputime(), walltime() R = PolynomialRing(GF(p), 'X') # If no polynomials are given, we compute both of them randomly. if f is None: f = R.irreducible_element(n, algorithm='random') if g is None: g = R.irreducible_element(n, algorithm='random') while f == g: g = R.irreductible_element(n, algorithm='random') # We compute two fields of cardinality p^n and two primitive m-rooth rootmf, rootmg, F, G = find_mroots_and_fields(p, n, m, f, g) # The matrixes will contain the coefficients of rootmf and rootmg in the # basis x^i and y^i respectively A = matrix(GF(p), n, n) B = matrix(GF(p), n, n) for i in range(n): A[i,:] = (rootmf**i).vector() # Failsafe, but probably outdated try: Ainv = A.inverse() except ZeroDivisionError: print 'erreur' return A # We will try to find the power s such as phi(rootmf) = rootmg^s, since # rootmg and rootmf are both primitive mrooth it is bound to happen # since the multiplicative group is cyclic. s = 1 while s <= m : for i in range(n): B[i,:] = ((rootmg**s)**i).vector() # This will be the isomorphism's matrix C = Ainv*B v = C[1,:] # The second line correponds to the image of x res = G(v[0]) # I realized that you could try to find if the image of # rootmf is also a zero of the minimal polynomial of rootmf # but it would force us to compute yet another minimal polynomials. # Instead, if you find that the image of x is a root of f, # then you win! if f(res) == 0: print 'CPU %s, Wall %s' % (cputime(c), walltime(w)) # Some of what is returned is probably useless and only here # for testing purposes. return (res, C, rootmf, rootmg, s, f, F, G) s = s + 1 print 'No isomorphism found, check your m.' return 1
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_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, structure=None): """ 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 a ``structure`` attribute for compatibility with :class:`~sage.categories.pushout.AlgebraicExtensionFunctor` but we ignore it as it is 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() 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. 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 # 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_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
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
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.rings.arith 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 # This check should be replaced by order.is_prime_power() # if Trac #16878 is fixed. elif sage.rings.arith.is_prime_power(order): if not names is None: name = names 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