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2019-01-31 09:04:37 +0100 asked a question x coordinate Of an Elliptic Curve point

I am defining an Elliptic curve E and then taking a random point P over E. now I want to print the x coordinate of the elliptic curve. How to do that ?

F.<z>=GF(2^11, modulus= conway_polynomial(2,11))
j= F.random_element()
E= EllipticCurve_from_j(j)
P= E.random_point()

Now I want to define w to be the x co-ordinate of P.

2019-01-26 16:06:11 +0100 commented answer Declaring symbols in a Field

thank you very much. Now it's working perfectly.

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2019-01-26 04:08:29 +0100 commented answer Declaring symbols in a Field

I think it is the latest version. i.e. 8.6 in macOs 10.13.6

sage: version()
        'SageMath version 8.6, Release Date: 2019-01-15'

in fact I checked in Cocalc and it has the same errors. Now I am really confused. :(

Your code if I can get it to work is exactly what I need

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2019-01-25 10:16:46 +0100 commented answer Declaring symbols in a Field

using your code I am getting this :

sage: R.<a1,b1,c1,a2,b2,c2> = BooleanPolynomialRing()
sage: S.<s> = PolynomialRing(R)
sage: T.<t> = S.quotient(conway_polynomial(2,3).change_variable_name('s'))
sage: X = a1 + b1*t + c1*t^2
sage: Y = a2 + b2*t + c2*t^2
sage: f = X^2*Y^2 - (X^2+(1+t)*Y^2)
sage: R.ideal(list(f)).variety()
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-9-9d89a195f359> in <module>()
----> 1 R.ideal(list(f)).variety()

TypeError: 'QuotientRing_generic_with_category.element_class' object is not iterable
2019-01-25 07:26:22 +0100 commented answer Declaring symbols in a Field

Thank you for the solution. But I don't understand what the T definition doing actually. if with the same X you compute T(X)^10 the output is c1t^20 + b1c1t^18 + a1c1t^16 + b1c1t^12 + b1t^10 + a1b1t^8 + a1c1t^4 + a1b1t^2 + a1, whereas according to my question it should come back to a poly of degree 2 in t as modulo the conway_polynomial (2,3).

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2019-01-23 15:20:27 +0100 asked a question Declaring symbols in a Field

Lets define a field F.<t> = GF(2^n) now i want to define a variable points of the form x1+x2t+...+xnt^(n-1) and then solving equation with this by comparing coefficients of t^i.

Now I am defining R= PolynomialRing(ZZ,'x',n) c=R.gens() R=R.quotient_ring([c[i]^2-c[i] for i in range(0,n)]) then i get n variables but if I write x= sum(c[i]*t^i for i in (0,n)) I get the parent of x is R. and I am unable to collect the coefficients of t^i. after defining y and z in the same way. if I do X=x+y+z then I am getting the value as a Ring element with monomials in xi's and coefficients in F as X in a Ring element.

Can anyone suggest any way to get the results as f1+f2t+f3t^2+...+fn*t^(n-1) and then collect fi's where fi's are functions in the variables xi's.

suppose I want to solve for X,Y in GF(2^3) with X^2Y^2= X^2+(1+t)Y^2

sage: F.<t>=GF(2^3)

sage: R.<a1,a2,b1,b2,c1,c2>= PolynomialRing(ZZ)

sage: R.<a1,a2,b1,b2,c1,c2>= R.quotient_ring([a1^2-a1,a2^2-a2,b1^2-b1,b2^2-b2,c1^2-c1,c2^2-c2])

sage: X=R(a1)+R(b1)t+R(c1)t^2

sage: Y=R(a2)+R(b2)t+R(c2)t^2

sage: X^2*Y^2

Now i want to store this output as A+Bt+Ct^2(how to do this? )

then I do:

sage: X^2+(1+t)*Y^2

and I want to store this as E+Ft+Gt^2(is it possible to do?)

finally I want to solve for ai's and bi's from the eqns A=E, B=F,C=G,ai^2=ai,bi^2=bi,ci^2=ci for i=1,2