SageMath: defining class of functions on Elliptic Curves
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
The example as given doesn't give an error. If I execute fADD_(P,Q,x,y), I get an error because
yis not defined. The error you describe suggests you may have executedvar('y')before. I don't know whatyshould be. Perhaps something like the code below?