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.Mon, 15 Jun 2020 17:14:14 +0200I need to represent the Elliptic curve point in projective coordinate with Z coordinate has value not normalized 1https://ask.sagemath.org/question/51120/i-need-to-represent-the-elliptic-curve-point-in-projective-coordinate-with-z-coordinate-has-value-not-normalized-1/the points over elliptic curve defined over finite field always displayed in projective coordinates (X:Y:Z) with normalised Z component ( always Z = 1). i need points to be in normal projective coordinates ( Z component not always 1 ) Tue, 28 Apr 2020 20:17:53 +0200https://ask.sagemath.org/question/51120/i-need-to-represent-the-elliptic-curve-point-in-projective-coordinate-with-z-coordinate-has-value-not-normalized-1/Comment by slelievre for <p>the points over elliptic curve defined over finite field always displayed in projective coordinates (X:Y:Z) with normalised Z component ( always Z = 1). i need points to be in normal projective coordinates ( Z component not always 1 ) </p>
https://ask.sagemath.org/question/51120/i-need-to-represent-the-elliptic-curve-point-in-projective-coordinate-with-z-coordinate-has-value-not-normalized-1/?comment=51147#post-id-51147Example code that can be copy-pasted in a fresh Sage session to illustrate the problem
helps others get started on answering a question, and therefore increases the chances
and the expected speed of an answer. Please give us something to play with!Wed, 29 Apr 2020 18:31:39 +0200https://ask.sagemath.org/question/51120/i-need-to-represent-the-elliptic-curve-point-in-projective-coordinate-with-z-coordinate-has-value-not-normalized-1/?comment=51147#post-id-51147Answer by dan_fulea for <p>the points over elliptic curve defined over finite field always displayed in projective coordinates (X:Y:Z) with normalised Z component ( always Z = 1). i need points to be in normal projective coordinates ( Z component not always 1 ) </p>
https://ask.sagemath.org/question/51120/i-need-to-represent-the-elliptic-curve-point-in-projective-coordinate-with-z-coordinate-has-value-not-normalized-1/?answer=52016#post-id-52016It is hard to extract (exactly the intentioned) question, but i will try to cover all nuances.
First of all, not all points are shown with a $Z$-normalized component, as the following example is showing it.
sage: E = EllipticCurve( GF(7), [3, 4] )
sage: for P in E:
....: print(P)
....:
(0 : 1 : 0)
(0 : 2 : 1)
(0 : 5 : 1)
(1 : 1 : 1)
(1 : 6 : 1)
(2 : 2 : 1)
(2 : 5 : 1)
(5 : 2 : 1)
(5 : 5 : 1)
(6 : 0 : 1)
And indeed, all **but one** points have a $Z$-component normalized to one, but the neutral point $O$ is shown as $(0:1:0)$. For all points we can initialize them using the $(x,y)$ notation (if the $Z$-component does not vanish), for instance:
sage: E.point( [2, 5] )
(2 : 5 : 1)
but also using "projective components" (in a list), for instance for the above point:
sage: E.point( [2, 5] )
(2 : 5 : 1)
sage: E.point( [2, 5, 1] )
(2 : 5 : 1)
sage: E.point( [4, 10, 2] )
(2 : 5 : 1)
sage: E.point( [6, 15, 3] )
(2 : 5 : 1)
sage: E.point( [-1, 1, 3] )
(2 : 5 : 1)
et caetera. The "infinity point" is
sage: E.point(0)
(0 : 1 : 0)
sage: E.point( [0, 1, 0] )
(0 : 1 : 0)
sage: E.point( [0, 2, 0] )
(0 : 1 : 0)
so we can introduce it in either of the above ways.
Mon, 15 Jun 2020 17:14:14 +0200https://ask.sagemath.org/question/51120/i-need-to-represent-the-elliptic-curve-point-in-projective-coordinate-with-z-coordinate-has-value-not-normalized-1/?answer=52016#post-id-52016