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/pairings/beginners/5-3-1-TateWeilMiller.txt
EDIT: my question hasn't attracted much interest so i'm going to give a more concrete example:
# this is the beginning of the code related to the example in Costello
q=47
F = GF(q)
R.<x> = F[] ; R
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 is a divisor of 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):
lamb_da=(Q[1]-P[1])/(Q[0]-P[0])
c =P[1]-lamb_da*P[0]
l =(y-(lamb_da*x+c))
v =(x-(lamb_da^2-P[0]-P[1]))
return (l/v)
fADD_(P,Q,x,y) will return an error while i would have liked it to return a rational function in x,y
--------------------------------------------------------------------------- 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.valueKeyError: (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_daP[Integer(0)] ----> 4 l =(y-(lamb_dax+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
