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2017-11-29 03:25:56 -0600 | commented question | Multiplication of elements of tower fields f1 has to be a R16 element, and A a R4 element. I build up a small, but ugly, work around. I take every A and put it in a List. Then, in the end, I return that list and convert each element to a Poly=PolynomialRing(R) element, where f1 is also an PolynomialRing(R) element. Then I do the multiplication over that PolynomialRing and do a "lazy reduction" in the end, by "Poly(R16(List[pos]))". From this position, I'm able to print via ".list()" any coefficient in Hex. |

2017-11-29 02:31:10 -0600 | asked a question | Multiplication of elements of tower fields Let I need to compute This element stays allways in R4. But the result i expect, is an element of R16. So: How can I tell sage to use the R16 multiplication instead? I need a full degree 15 polynomial. If it is possible, in one variable (w, since w^4=v). Furthermore: How can I tell Sage to print any coefficient in Hex? |

2017-11-27 03:26:34 -0600 | commented question | Defining a subgroup of elliptic curves with specific characteristics My Question is actually obsolet. I will add an answere and some miss-thoughts in the next days. |

2017-11-26 14:52:29 -0600 | commented question | Defining a subgroup of elliptic curves with specific characteristics Ok. Thinking about you mentioned behaviour delivers, that this can't work. I'll add a more precise question. |

2017-11-26 12:30:01 -0600 | commented question | Defining a subgroup of elliptic curves with specific characteristics I'm on generating an example from now on. |

2017-11-25 10:11:03 -0600 | commented question | Defining a subgroup of elliptic curves with specific characteristics Ok. You got it right. I might produce an example in the next hours. For instance the parameters "p=13" and "r=5" should do it. Consider $E':\ y^2=x^3+ (2^{-1/4})x$ over $\mathbb F_{p^4}$. This curve becomes isomorphic over $\mathbb F_{p^{16}}$ to $E:\ y^2=x^3+x$, where $E(\mathbb F_p)[r]=E(\mathbb F_{p^{16}})[r]\cap {P\in E:\ \pi (P)= \mathcal O }$. (Twist) |

2017-11-23 05:57:14 -0600 | commented question | Defining a subgroup of elliptic curves with specific characteristics Ok, I do. :) $\pi$ is the p-Frobenius, that means: $\pi(P)=(x(P)^p, y(P)^p)$ and the prime is around 340 bits of size. You can find that in the edit also. |

2017-11-22 07:16:18 -0600 | asked a question | Defining a subgroup of elliptic curves with specific characteristics Hey, is there a way, to define a subgroup of an elliptic curve with two or more characteristics? I would like to take an elliptic curve over a finite field of order p and $p^4$, define the r-torsion subgroup (where $r$ is a prime, too) and reduce those to the set of points, which also lays in the Frobenius-eigenspace. For example: Well, it is easy to find a point on $Q\in E4$, such that $r*Q = (0:1:0)$, use , but checking, if $( x(Q)^p, y(Q)^p )=\pi(Q) = pQ$ is hard. I only need one point of that group at all, but my $p$ is even larger (~340 bits), so brute-forcing would be an option, if I could start it 6-12 month ago :) Furthermore, if I concider to evaluate the secant or tangent on E and let me return a point on that curve, it will have projective coordinates, with $z(P)\neq1$. Shall I apply $\pi$ to all three coordinates? |

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2017-11-17 06:47:06 -0600 | commented answer | How to defining a twist on an elliptic curve Wow nice, thank you. Yesterday, after some hours of wondering, why this does not work or even produces different results, as I thought to get, I saw, that I missunderstood twists. They are becoming isomorphic over the extension field, not through the different fields ($F_k:=\mathbb F_{p^k}$) $F_4$ and $F_{16}$. So the r-torsion point I found worked over $F_{16}$, but now over $F_4$. |

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2017-11-16 05:17:16 -0600 | asked a question | How to defining a twist on an elliptic curve Hey, I would like to do map points of the ellitptic curve $E(\mathbb F_{p^{k}})$ to its twist. I am able to define the twist on a mathematical way, but it returns always errors. I will give you a M(n)WE: Aim: Get a generator of a r-torsion subgroup of E, lift that to E16 and twist it down to E4, to get a r-torsion subgroup-generator of E4. ## MWE## Mathematical definition:$E(\mathbb F_{p^{16}}): y^2=x^3+x$ and its quartic twist $E'(\mathbb F_{p^4}): y^2=x^3+2^{-1/4}x$. The point mapping is defined as $\psi: E\to E', (x,y,z)\mapsto (2^{-1/2}x, 2^{-3/4}y, z)$. ## Define the fields and curves ## Defining the point and pre-computations ## The not working twist Sage is able to compute the quartic twist of its own, but I do not recieve the right twist, that I had computed by hand ( $E'$ ). Using |