ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Wed, 25 Nov 2020 13:40:37 +0100How define Twisted Edwards elliptic curve ?https://ask.sagemath.org/question/54383/how-define-twisted-edwards-elliptic-curve/In Sage, `EllipticCurve` takes 2 or 5 parameters.
For 5 parameters, `EllipticCurve([a, b, c, d, e])` means:
`y^2 + a*x*y + c*y = x^3 + b*x^2 + d*x + e`.
The curve Ed25519 (for example for Monero) is defined by
$$
-x^2+y^2=1-\frac{121665}{121666}x^2y^2
$$
This equation has a term in $x^2y^2$ which does not exist in Sage `EllipticCurve` parameters.
What does "twisted" mean?
How to define $\frac{121665}{121666}$ ?
I must multiply 121665 but inverted 121666 in field? Field is with prime p=$2^{255}-19$.AndrWed, 25 Nov 2020 13:40:37 +0100https://ask.sagemath.org/question/54383/Twists of an Elliptic Curve over a Finite Field (secp256k1)https://ask.sagemath.org/question/42306/twists-of-an-elliptic-curve-over-a-finite-field-secp256k1/ I would like to calculate the twists of secp256k1 up to isomorphism (find one representative for the quadratic twist, and so on). Therefore I tried the function <br/>
E.quadratic_twist(d) and E.sextic_twist(d) <br/>
with E = EllipticCurve(GF(115792089237316195423570985008687907853269984665640564039457584007908834671663), [0,0,0,0,7]).
<br/>
The problem is that I don't understand what the variable $d$ does in this function. <br/>
I tried different $d$ and found a quadratic twist with $d=3$ $E' : y^2 = x^3 + 12096 = x^3 + 7 \cdot (12)^3$ and tested that $12$ is square-free in $\mathbb{F}_q$, and two sextic twists with $d=3,7$: $E^{d_6}_1: y^2 = x^3 + 7\cdot 139968$ and $E^{d_6}_2: y^2 = x^3 + 7\cdot 326592$ with $139968$ and $326592$ square- and cube-free in $\mathbb{F}_q$. <br/>
<br/>
But my method is rather brute-force, since I don't really understand the function, and I am also not sure how many different sextic twists sexp256k1 has (up to isomorphism). I think it is two, but I am not sure. <br/>
<br/>
My next question is, if there is a way to calculate the cubic twist with a similar function?
SoftyThu, 10 May 2018 16:55:45 +0200https://ask.sagemath.org/question/42306/How do I quickly setup a remote Sage server?https://ask.sagemath.org/question/7644/how-do-i-quickly-setup-a-remote-sage-server/I ssh'd to a remote machine on my local LAN, ran "./sage -n"; but I can't login to port 8000 remotely with https or http. Does twisted by default disallow remote connections, or are there likely platform-by-platform firewall issues? What's the quickest and closest thing to a zero-configuration setup? Like most casual users, I just "want it to work" -- minimal setup fuss. This isn't a complaint, but I think a common question.ccanoncFri, 27 Aug 2010 21:10:22 +0200https://ask.sagemath.org/question/7644/