fagui's profile - activity

Hi i'm a beginner with Sage, and i have some trouble with types. It looks like the cause of my pain is trying to use numpy together with Sagemath, is it possible to use Sagemath without numpy?

Here is a small example of an error that i encounter, when trying to do simple modular calculations in GF(7).

I use the following code:

import numpy as np

a=[2,4,6,1,3,5]
b=[1,1,1,1,1,1]
p=7
n=len(a)
L=1
c=np.zeros(n,'int')
Li_ai=np.zeros(n)
for i in range(n):
    L=(x-a[i])*L
for i in range(n):
    Li=L/(x-a[i])
    t=(Li(x=a[i])%p)
    Li_ai[i]=t
Li_ai=Li_ai.astype('int')
tmp=Li_ai.prod()%p
Li_ai=vector(Li_ai)
for i in range(n):
    c[i]=tmp * (Li_ai[i]) * b[i].inverse_mod(p) %p
c=vector(c)
c
y=np.int(tmp)
y.inverse_mod(p)

and I get the following error:

Traceback (most recent call last)
...
AttributeError: 'int' object has no attribute 'inverse_mod'

Hello

I'm following a 101 algebra course, and for example, I would like to factor a polynomial on a finite field like F_9 (F_9 == ZZ/9ZZ is a field because 9 is a power of a prime number, 3)

R = PolynomialRing(GF(9),'x')
x = R.gen()
f = x^4+x^2-1
f in R
f.factor()

i get an error message

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_68.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("UiA9IFBvbHlub21pYWxSaW5nKEdGKDkpLCd4JykKeCA9IFIuZ2VuKCkKZiA9IHheNCt4XjItMQpmIGluIFI="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>

  File "/tmp/tmpYgKq_W/___code___.py", line 3, in <module>
    R = PolynomialRing(GF(_sage_const_9 ),'x')
  File "factory.pyx", line 364, in sage.structure.factory.UniqueFactory.__call__ (build/cythonized/sage/structure/factory.c:1244)
  File "/home/sage/sage-6.3/local/lib/python2.7/site-packages/sage/rings/finite_rings/constructor.py", line 414, in create_key_and_extra_args
    raise ValueError("parameter 'conway' is required if no name given")
ValueError: parameter 'conway' is required if no name given

i'm running sage 6.3 notebook on windows through Oracle VM winbox. I'm a totally new user, and i've looked at the tutorial and the forum but couldn't find any example or reason why this would not work. thank you for your help.

In SageMath,

I would like to manipulate rational functions on elliptic curves (defined on finite fields). For example, for $P = (x,y)$ on some curve $E$

$$f = x+y-12$$ $$g = \frac{x+y-3}{(x-3)^2} $$ etc.

Is there a natural class?

I am looking to make a toy example with pairings, so I need to define stuff like $$P \rightarrow f_P$$ where $$f_P:Q \rightarrow f_P(Q)$$ is a function

I can't see how to do that, and I'm able to make computations if I define

def f (P,Q):
   ....

but I can only compute the values taken by the function $f_P$, I cannot "see" the function $f_P$. Basically I'm trying as an exercise to re-write the following Magma code to SageMath:

http://www.craigcostello.com.au/pairi...

EDIT

My question hasn't attracted much interest so let me give a more concrete example.

Define the following:

# Code related to the example in Costello
q = 47
F = GF(q)
R.<x> = F[]
F4.<u> = F.extension(x^4 - 4*x^2 + 5)
a = 21
b = 15
E = EllipticCurve(F4, [a, b])
r = 17
k = 4
(q^4 - 1) % r  # r = 17 divides q^4 - 1 = 47^4 - 1
P = E([45, 23])
P.order()
h = E.cardinality() / r^2
O = E(0)
Q = E([5*u^3 + 37*u + 13, 7*u^3 + 45*u^2 + 10*u + 7])

def fADD_(P, Q, x, y):
    lamda = (Q[1] - P[1]) / (Q[0] - P[0])
    c = P[1] - lamda * P[0]
    l = (y - (lamda * x + c))
    v = (x - (lamda^2 - P[0] - P[1]))
    return l / v

I would have hoped for fADD_(P, Q, x, y) to return a rational function in x, y.

Instead it raises the following error:

---------------------------------------------------------------------------
KeyError                                  Traceback (most recent call last)
/Applications/SageMath-9.1.app/Contents/Resources/sage/local/lib/python3.7/site-packages/sage/structure/coerce.pyx
in sage.structure.coerce.CoercionModel.bin_op
(build/cythonized/sage/structure/coerce.c:9946)()    1195         try:
-> 1196             action = self._action_maps.get(xp, yp, op)    1197         except KeyError:

/Applications/SageMath-9.1.app/Contents/Resources/sage/local/lib/python3.7/site-packages/sage/structure/coerce_dict.pyx
in sage.structure.coerce_dict.TripleDict.get
(build/cythonized/sage/structure/coerce_dict.c:7917)()    1327        
if not valid(cursor.key_id1):
-> 1328             raise KeyError((k1, k2, k3))    1329         value = <object>cursor.value

KeyError: (Finite Field in u of size 47^4, Symbolic Ring, <built-in
function mul>)

During handling of the above exception, another exception occurred:

TypeError                                 Traceback (most recent call
last) <ipython-input-54-7441affd269c> in <module>()
----> 1 fADD_(P,Q,x,y)

<ipython-input-48-20f93738e654> in fADD_(P, Q, x, y)
      2     lamb_da=(Q[Integer(1)]-P[Integer(1)])/(Q[Integer(0)]-P[Integer(0)])
      3     c =P[Integer(1)]-lamb_da*P[Integer(0)]
----> 4     l =(y-(lamb_da*x+c))
      5     v =(x-(lamb_da**Integer(2)-P[Integer(0)]-P[Integer(1)]))
      6     return (l/v)

/Applications/SageMath-9.1.app/Contents/Resources/sage/local/lib/python3.7/site-packages/sage/structure/element.pyx
in sage.structure.element.Element.__mul__
(build/cythonized/sage/structure/element.c:12034)()    1515           
return (<Element>left)._mul_(right)    1516         if
BOTH_ARE_ELEMENT(cl):
-> 1517             return coercion_model.bin_op(left, right, mul)    1518     1519         cdef long value

/Applications/SageMath-9.1.app/Contents/Resources/sage/local/lib/python3.7/site-packages/sage/structure/coerce.pyx
in sage.structure.coerce.CoercionModel.bin_op
(build/cythonized/sage/structure/coerce.c:9996)()    1196            
action = self._action_maps.get(xp, yp, op)    1197         except
KeyError:
-> 1198             action = self.get_action(xp, yp, op, x, y)    1199         if action is not None:    1200             if
(<Action>action)._is_left:

/Applications/SageMath-9.1.app/Contents/Resources/sage/local/lib/python3.7/site-packages/sage/structure/coerce.pyx
in sage.structure.coerce.CoercionModel.get_action
(build/cythonized/sage/structure/coerce.c:16783)()    1725        
except KeyError:    1726             pass
-> 1727         action = self.discover_action(R, S, op, r, s)    1728         action = self.verify_action(action, R, S, op)    1729        
self._action_maps.set(R, S, op, action)

/Applications/SageMath-9.1.app/Contents/Resources/sage/local/lib/python3.7/site-packages/sage/structure/coerce.pyx
in sage.structure.coerce.CoercionModel.discover_action
(build/cythonized/sage/structure/coerce.c:18201)()    1856         """
1857         if isinstance(R, Parent):
-> 1858             action = (<Parent>R).get_action(S, op, True, r, s)    1859             if action is not None:    1860                 return
action

/Applications/SageMath-9.1.app/Contents/Resources/sage/local/lib/python3.7/site-packages/sage/structure/parent.pyx
in sage.structure.parent.Parent.get_action
(build/cythonized/sage/structure/parent.c:19901)()    2475        
action = self._get_action_(S, op, self_on_left)    2476         if
action is None:
-> 2477             action = self.discover_action(S, op, self_on_left, self_el, S_el)    2478     2479         if action is not None:

/Applications/SageMath-9.1.app/Contents/Resources/sage/local/lib/python3.7/site-packages/sage/structure/parent.pyx
in sage.structure.parent.Parent.discover_action
(build/cythonized/sage/structure/parent.c:20878)()    2554            
# detect actions defined by _rmul_, _lmul_, _act_on_, and _acted_upon_ methods    2555                 from .coerce_actions import
detect_element_action
-> 2556                 action = detect_element_action(self, S, self_on_left, self_el, S_el)    2557                 if action is not
None:    2558                     return action

/Applications/SageMath-9.1.app/Contents/Resources/sage/local/lib/python3.7/site-packages/sage/structure/coerce_actions.pyx
in sage.structure.coerce_actions.detect_element_action
(build/cythonized/sage/structure/coerce_actions.c:5026)()
    215     if isinstance(x, ModuleElement) and isinstance(y, Element):
    216         try:
--> 217             return (RightModuleAction if X_on_left else LeftModuleAction)(Y, X, y, x)
    218         except CoercionException as msg:
    219             _record_exception()

/Applications/SageMath-9.1.app/Contents/Resources/sage/local/lib/python3.7/site-packages/sage/structure/coerce_actions.pyx
in sage.structure.coerce_actions.ModuleAction.__init__
(build/cythonized/sage/structure/coerce_actions.c:6778)()
    361         if not isinstance(g, Element) or not isinstance(a, ModuleElement):
    362             raise CoercionException("not an Element acting on a ModuleElement")
--> 363         res = self.act(g, a)
    364         if parent(res) is not the_set:
    365             # In particular we will raise an error if res is None

/Applications/SageMath-9.1.app/Contents/Resources/sage/local/lib/python3.7/site-packages/sage/categories/action.pyx
in sage.categories.action.Action.act
(build/cythonized/sage/categories/action.c:4115)()
    213             5*x
    214         """
--> 215         return self._act_convert(g, x)
    216 
    217     def __invert__(self):

/Applications/SageMath-9.1.app/Contents/Resources/sage/local/lib/python3.7/site-packages/sage/categories/action.pyx
in sage.categories.action.Action._act_convert
(build/cythonized/sage/categories/action.c:3759)()
    169         if parent(x) is not U:
    170             x = U(x)
--> 171         return self._act_(g, x)
    172 
    173     cpdef _act_(self, g, x):

/Applications/SageMath-9.1.app/Contents/Resources/sage/local/lib/python3.7/site-packages/sage/structure/coerce_actions.pyx
in sage.structure.coerce_actions.RightModuleAction._act_
(build/cythonized/sage/structure/coerce_actions.c:8600)()
    629             g = <Element?>self.connecting._call_(g)
    630         if self.extended_base is not None:
--> 631             a = <ModuleElement?>self.extended_base(a)
    632         return (<ModuleElement>a)._lmul_(<Element>g)  # a * g
    633 

/Applications/SageMath-9.1.app/Contents/Resources/sage/local/lib/python3.7/site-packages/sage/structure/parent.pyx
in sage.structure.parent.Parent.__call__
(build/cythonized/sage/structure/parent.c:9218)()
    898         if mor is not None:
    899             if no_extra_args:
--> 900                 return mor._call_(x)
    901             else:
    902                 return mor._call_with_args(x, args, kwds)

/Applications/SageMath-9.1.app/Contents/Resources/sage/local/lib/python3.7/site-packages/sage/structure/coerce_maps.pyx
in sage.structure.coerce_maps.DefaultConvertMap_unique._call_
(build/cythonized/sage/structure/coerce_maps.c:4556)()
    159                 print(type(C), C)
    160                 print(type(C._element_constructor), C._element_constructor)
--> 161             raise
    162 
    163     cpdef Element _call_with_args(self, x, args=(), kwds={}):

/Applications/SageMath-9.1.app/Contents/Resources/sage/local/lib/python3.7/site-packages/sage/structure/coerce_maps.pyx
in sage.structure.coerce_maps.DefaultConvertMap_unique._call_
(build/cythonized/sage/structure/coerce_maps.c:4448)()
    154         cdef Parent C = self._codomain
    155         try:
--> 156             return C._element_constructor(x)
    157         except Exception:
    158             if print_warnings:

/Applications/SageMath-9.1.app/Contents/Resources/sage/local/lib/python3.7/site-packages/sage/symbolic/ring.pyx
in sage.symbolic.ring.SymbolicRing._element_constructor_
(build/cythonized/sage/symbolic/ring.cpp:6648)()
    377         elif isinstance(x, (RingElement, Matrix)):
    378             if x.parent().characteristic():
--> 379                 raise TypeError('positive characteristic not allowed in symbolic computations')
    380             exp = x
    381         elif isinstance(x, Factorization):

TypeError: positive characteristic not allowed in symbolic
computations

I'm trying this code base on the book "Adventures in Group Theory" from D.Joyner

MS=MatrixSpace(GF(2),3,3) 
A=MS([0,1,0,1,0,0,0,0,1]) 
B=MS([1,0,0,0,0,1,0,1,0])
G=MatrixGroup([A,B]) G1=SymmetricGroup(3)
imG=G._gap_().IsomorphismPermGroup().Image(G._gap_())
imG

Sym( [ 2 .. 4 ] )

gens=imG.GeneratorsOfGroup()
gens

[ (3,4), (2,3) ]

G2=PermutationGroup([(3,4),(2,3)])
G2.is_isomorphic(G1)

True

However, there seems to be a signature problem the following commands don't work: (the first one was from the book, I tried myself the second one)

G2=PermutationGroup(gens)

Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_92.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -- coding: utf-8 --\n" + _support_.preparse_worksheet_cell(base64.b64decode("RzI9UGVybXV0YXRpb25Hcm91cChnZW5zKQ=="),globals())+"\n"); execfile(os.path.abspath("___code___.py")) File "", line 1, in <module>

File "/private/var/folders/gm/z065gk616xg6g0xgn4c7_bvc0000gn/T/tmpHRYMWK/___code___.py", line 2, in <module> exec compile(u'G2=PermutationGroup(gens) File "", line 1, in <module>

File "/Applications/SageMath/local/lib/python2.7/site-packages/sage/groups/perm_gps/permgroup.py", line 343, in PermutationGroup raise TypeError("gens must be a tuple, list, or GapElement") TypeError: gens must be a tuple, list, or GapElement

G2=PermutationGroup(list(gens))

Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_79.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -- coding: utf-8 --\n" + _support_.preparse_worksheet_cell(base64.b64decode("RzI9UGVybXV0YXRpb25Hcm91cChsaXN0KGdlbnMpKQ=="),globals())+"\n"); execfile(os.path.abspath("___code___.py")) File "", line 1, in <module>

File "/private/var/folders/gm/z065gk616xg6g0xgn4c7_bvc0000gn/T/tmpOxJAnr/___code___.py", line 2, in <module> exec compile(u'G2=PermutationGroup(list(gens)) File "", line 1, in <module>

File "/Applications/SageMath/local/lib/python2.7/site-packages/sage/groups/perm_gps/permgroup.py", line 345, in PermutationGroup canonicalize=canonicalize, category=category) File "/Applications/SageMath/local/lib/python2.7/site-packages/sage/groups/perm_gps/permgroup.py", line 419, in __init__ for cycle in x: TypeError: 'sage.libs.gap.element.GapElement_Permutation' object is not iterable

I cannot make much of the documentation. Thanks.

thank you very much, i need some time to digest it all, because i'm a beginner/amateur in this topic. i don't really understand what f and g are.

i think they should be well understood to some extent. there are the basis of some powerful popular methods for integer factorization or proving that a large number is prime for example.

do you see a way around that ?

An Elliptic curve is the union of its affine part Eaff(Z/nZ)={ [x,y,1] in P2(Z/nZ) such that y2 =x3+ax+b } and the point at infinity O = [0,1,0]

P2(Z/nZ) is the projective plane:

T = { (x,y,z) in (Z/nZ)^3 such that gcd(x,y,z,n) = 1}

P2(Z/nZ) = (T / ~) where ~ is the equivalence relation defined by (x,y,z) ~ (x0,y0,z0) iff there is an INVERSIBLE element u in Z/nZ ⇤ such that (x,y,z) = u(x0,y0,z0).

I found those doc pages

http://fe.math.kobe-u.ac.jp/icms2010-...

http://doc.sagemath.org/html/en/refer...

I think to define an Elliptic Curve here with

q=10
E1 = EllipticCurve(Zmod(q),[0,1])
E1

Elliptic Curve defined by y^2 = x^3 + 1 over Ring of integers modulo 10

but

E1.points()

is not working. is there a method to get all the points of this Elliptic Curve ? its cardinality etc ????

if E an elliptic is defined on K, then E[n] is a subgroup of an Elliptic curve defined on an algebraic closure of K, its the set of points P of this curve whose order is dividing n

G = E.torsion_subgroup() gives the torsion subgroup in E(K), but is there a method giving the E[n] ?

For example, for the Elliptic Curve y^2=x^3-2 defined on Q

if we denote by alpha the cubic root of 2 and J a complex cubic root of unity, and O the point at infinity

E[2] = {O,(alpha,0), (alpha * J,0),(alpha * J^2,0)}

I tried the following

E = EllipticCurve(CC,[0,-2])

O = E(0)

O.division_points(2)

and got [pretty close but not the right result, and not in an algebraic form anyway]

 [(0.000000000000000 : 1.00000000000000 : 0.000000000000000),
 (1.25992104989487 : 0.000000000000000 : 1.00000000000000)]

thanks