Transformation between distributions in time–frequency analysis: Difference between revisions

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{{Refimprove|date=January 2010}}
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{{Technical|date=January 2010}}
[[Image:Twisted Edwards curve.svg|300px|right|thumb|A Twisted Edwards curve of equation <math>10x^2+y^2=1+6x^2y^2</math>]]
 
In [[algebraic geometry]], the '''Twisted Edwards curves''' are plane models of [[elliptic curve]]s, a generalisation of [[Edwards curve]]s introduced by [[Daniel J. Bernstein|Bernstein]] et al. (2007) and named after [[Harold Edwards (mathematician)|Harold M. Edwards]].  Each [[twists of curves|twisted]] Edwards curve (as the name suggests) is a [[Twists of curves|twist]] of an [[Edwards curve]].
A twisted Edwards curve E <sub>E</sub><sub>,</sub><sub>a</sub><sub>,</sub><sub>d</sub> over a [[field (mathematics)|field]] ''K'' which does not have 2=0 is an [[affine]] plane curve defined by the equation:
 
:E <sub>E</sub><sub>,</sub><sub>a</sub><sub>,</sub><sub>d</sub>: <math>a x^2+y^2= 1+dx^2y^2</math>
 
where ''a'', ''d'' are distinct non-zero elements of ''K''. An Edwards curve is a twisted Edwards curve with ''a'' = 1.
 
Every twisted Edward curve is [[birationally equivalent]] to an elliptic curve in [[Montgomery curve|Montgomery form]] and vice versa.
 
==Group law==
 
As for all elliptic curves, also for the Twisted Edwards curve, it is possible to do some operations between its points, such as adding two of them or doubling (or tripling) one. The results of these operations are always points that belong to the curve itself. In the following sections some formulas are given to obtain the coordinates of a point resulted from an addition between two other points (addition), or the coordinates of point resulted from a doubling of a single point on a curve.
 
===Addition on Twisted Edward curves===
Let ''K'' be a field with [[characteristic (algebra)|characteristic]] different from 2.
Let <math>(x_1,y_1)</math> and <math>(x_2,y_2)</math> be points on the Twisted Edward curve. The equation  of Twisted Edward curve is written as;
 
E<sub>E</sub><sub>,</sub><sub>a</sub><sub>,</sub><sub>d</sub>: <math>ax^2+y^2=1+dx^2y^2</math>.
 
The sum of these points <math>(x_1,y_1)    ,   (x_2,y_2)</math> on  E<sub>E</sub><sub>,</sub><sub>a</sub><sub>,</sub><sub>d</sub> is:
 
: <math>(x_1,y_1) + (x_2,y_2) = \left(\frac{x_1y_2+y_1x_2}{1+dx_1x_2y_1y_2} , \frac{y_1y_2-ax_1x_2}{1-dx_1x_2y_1y_2}\right)</math>
 
The neutral element is (0,1) and the negative of <math>(x_1,y_1)</math> is (<math>-x_1,y_1)</math>
 
These formulas also work for doubling. If ''a'' is a ''square'' in ''K'' and ''d'' is a ''non-square'' in&nbsp;''K'', these formulas are ''complete'': this means that they can be used for all pairs of points without exceptions; so they work for doubling as well, and neutral elements and negatives are accepted as inputs.<ref>Daniel J. Bernstein and Tanja Lange, ''Faster addition and doubling on elliptic curves''</ref>
 
'''Example of addition'''
 
Given the following Twisted Edwards curve with a=3 and d=2:
 
<math>3x^2 + y^2 = 1 + 2x^2y^2</math>
 
it is possible to add the points <math>P_1=(1,\sqrt{2})</math> and <math>P_2=(1,-\sqrt{2})</math> using the formula given above. The result is a point P<sub>3</sub> that has coordinates:
 
<math>x_3 = \frac{x_1y_2+y_1x_2}{1+dx_1x_2y_1y_2} = 0</math>
 
<math>y_3 = \frac{y_1y_2-ax_1x_2}{1-dx_1x_2y_1y_2} = -1</math>.
 
===Doubling on Twisted Edward curves===
''Doubling'' can be performed with exactly the same formula as addition.
Doubling of a point (x<sub>1</sub>,y<sub>1</sub>) on the curve E<sub>E</sub><sub>,</sub><sub>a</sub><sub>,</sub><sub>d</sub> is:
 
[2](''x''<sub>1</sub>,''y''<sub>1</sub>) = (''x''<sub>3</sub>,''y''<sub>3</sub>)
 
where
 
: <math>x_3= \frac{x_1y_1+y_1x_1}{1+dx_1x_1y_1y_1}=\frac{2x_1y_1}{ax_1^2+y_1^2}</math>
 
<!-- spacing for legibility -->
 
: <math>y_3= \frac{y_1y_1-ax_1x_1}{1-dx_1x_1y_1y_1}=\frac{y_1^2-ax_1^2}{2-ax_1^2-y_1^2}.</math>
 
'''Example of doubling'''
 
Considering the same twisted Edwards curve given in the previous example, with a=3 and d=2, it is possible to double the point <math>P_1=(1,\sqrt{2})</math>. The point 2P<sub>1</sub> obtained using the formula above has the following coordinates:
 
<math>x_3 = \frac{x_1y_1+y_1x_1}{1+dx_1x_1y_1y_1} = \frac{2\sqrt{2}}{5}</math>
 
<math>y_3 = \frac{y_1y_1-ax_1x_1}{1-dx_1x_1y_1y_1} = \frac{1}{3}.</math>
 
It is easy to see, with some little computations, that the point <math>P_3=(\frac{2\sqrt{2}}{5},\frac{1}{3})</math> belongs to the curve <math>3x^2 + y^2 = 1 + 2x^2y^2</math>.
 
==Extended coordinates==
 
There is another kind of coordinate system with which a point in the Twisted Edwards curves can be represented.
A point <math>(x,y,z) </math> on <math>ax^2+y^2= 1+dx^2y^2</math> is represented as X, Y, Z, T satisfying the following equations
x=X/Z, y=Y/Z, xy=T/Z.
 
The coordinates of the point (X:Y:Z:T) are called the '''extended Twisted Edward coordinates'''. The identity element is represented by (0:1:1:0). The negative of a point is (-X:Y:Z:-T).
 
===Inverted Twisted Edwards Coordinates===
 
The coordinates of the point <math>(X_1:Y_1:Z_1)</math>  are called the '''inverted twisted Edwards coordinates''' on the curve
<math>(X^2+aY^2)Z^2= X^2Y^2+dZ^4</math>with <math>X_1Y_1Z_1</math>≠0; this point to the affine one<math>(Z_1/X_1, Z_1/Y_1)</math> on E<sub>E</sub><sub>,</sub><sub>a</sub><sub>,d</sub>.
Bernstein and Lange introduced these inverted coordinates, for the case a=1 and observed that the coordinates save time in addition.
 
===Projective twisted Edward coordinates===
 
The equation for the Projective Twisted Edwards Curve is given as:
<math>(aX^2+Y^2)Z^2= Z^4+dX^2Y^2</math> For Z<sub>1</sub>≠0 the point (X<sub>1</sub>:Y<sub>1</sub>:Z<sub>1</sub>) represents the [[affine space|affine point]] (x<sub>1</sub>= X<sub>1</sub>/Z<sub>1</sub>, y<sub>1</sub>= Y<sub>1</sub>/Z<sub>1</sub>) on E<sub>E</sub><sub>,</sub><sub>a</sub><sub>,</sub><sub>d</sub>.
 
Expressing an elliptic curve in twisted Edwards form saves time in arithmetic, even when the same curve can be expressed in the Edwards form. To know more about the speeds of addition and doubling in projective coordinates on Edwards curves, standard coordinates on twisted Edward curves, inverted coordinates on Edwards curves and inverted coordinates on twisted Edwards curves refer to the table in:
 
http://hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html
 
==Addition in Projective Twisted curves==
 
The addition on a projective twisted Edwards curve is given by:
 
:(X<sub>3</sub>:Y<sub>3</sub>:Z<sub>3</sub>)= (X<sub>1</sub>:Y<sub>1</sub>:Z<sub>1</sub>)+(X<sub>2</sub>:Y<sub>2</sub>:Z<sub>2</sub>)
 
and it costs 10'''M'''ultiplications + 1'''S'''quaring + 2'''D'''(multiplication by the curve parameter ''d'') + 7 '''a'''dditions where the '''2D''' are one multiplication by ''a'' and one by ''d.''
 
;Algorithm
 
:A= Z<sub>1</sub>.Z<sub>2</sub>,
:B=A<sup>2</sup>
:C=X<sub>1</sub>.X<sub>2</sub>
:D=Y<sub>1</sub>.Y<sub>2</sub>
:E=dC.D
:F=B-E
:G=B+E
:X<sub>3</sub>= A.F((X<sub>1</sub>+Y<sub>1</sub>).(X<sub>2</sub>+Y<sub>2</sub>)-C-D)
:Y<sub>3</sub>=A.G.(D-aC)
:Z<sub>3</sub>=F.G<ref name = "Twisted Edwards Curves">D.J. Bernstein, P. Birkner, M. Joye, T. Lange, C. Peters, ''Twisted Edwards Curves''</ref>
 
===Doubling on projective twisted curves===
 
Doubling on the projective twisted curve is given by:
 
:(X<sub>3</sub>:Y<sub>3</sub>:Z<sub>3</sub>)= 2(X<sub>1</sub>:Y<sub>1</sub>:Z<sub>1</sub>)
 
This costs 3'''M'''ultiplications+4'''S'''quarings+1'''D'''+7'''a'''dditions where 1'''D''' is a multiplication by '''a'''
 
;Algorithm:
 
:B=(X<sub>1</sub>+Y<sub>1</sub>)<sup>2</sup>
:C= X<sub>1</sub><sup>2</sup>
:D=Y<sub>1</sub><sup>2</sup>
:E=aC
:F= E+D
:H=Z<sub>1</sub><sup>2</sup>
:J=F-2H
:X<sub>3</sub>=(B-C-D).J
:Y<sub>3</sub>=F.(E-D)
:Z<sub>3</sub>= F.J<ref name = "Twisted Edwards Curves"/>
 
==See also==
 
* [[EdDSA]]
* For more information about the running-time required in a specific case, see [[Table of costs of operations in elliptic curves]].
 
==Notes==
{{Reflist}}
 
==References==
 
* {{Citation
|author=Daniel J. Bernstein, Marc Joye, Tanja Lange, Peter Birkner, Christiane Peters
|title=Twisted Edwards Curves
|url=http://eprint.iacr.org/2008/013.pdf
}}
 
* {{Citation
|author=Huseyin Hisil, Kenneth Wong, Gary Carter, Ed Dawson.
|title=Twisted Edwards Curves revisited
|url=http://eprint.iacr.org/2008/522
}}
 
* {{Citation
|author=Daniel J. Bernstein, Tanja Lange, Peter Birkner, Christiane Peters
|title=ECM using Edward curves
|url=http://eprint.iacr.org/2008/016.pdf
}}
 
==External links==
* http://hyperelliptic.org/EFD/g1p/index.html
* http://hyperelliptic.org/EFD/g1p/auto-twisted.html
* The Ed25519 algorithm: http://ed25519.cr.yp.to/
 
{{DEFAULTSORT:Twisted Edwards Curve}}
[[Category:Elliptic curves]]
[[Category:Elliptic curve cryptography]]

Revision as of 19:52, 22 February 2014

Open world video games or sometimes referred to as sandbox are now gaining acceptance due to their non linear structure that helps the player simply roam the digitalized game environment with no apparent linear levels. The design provides the player the freedom to reach the game optimum, through formulation of his own techniques and strategies, and subsequent his own favorite game level sequence other than being compelled to adhere to the restricted linear game level sequence. Minecraft is one of the most used sandbox game that have a non linear game structure. It has three dimensional construction video games where the user is suppose to construct structures using textured cubes as the building blocks. Let's now take a close look at the details of this open world game and find out how, where it can be used and how to quickly get the free Minecraft premium gift code.



Minecraft as mentioned earlier is a sandbox building construction game. The game was initially conceptualized and developed by Markus Persson, his company (Mojang) is still improving on the features of this interactive game. The user or player of the game is required to build any imaginable structure of geometrical shape using textured blocks or cubes during the day. During the night, the monsters come out, and the structures need to have been built or erected before they emerge as a result of providing shelter against the monsters. The game was developed to be used on Java system and Java Applet. Since its inception in May, 2009, the game has turn into popular in the world of virtualization. This is because of its numerous features and the three dimensional impact.

The development of Minecraft has brought about substantial amounts of inspiration from other similar games such as Dungeon Keeper, Dwarf Fortress and Infiniminer. The Minecraft, however, can be played in two levels, thus classic and beta. The classic is the elementary level concerned with only building and construction. On the other hand, beta level is an advanced level where the player is exposed to attackers or enemies, the player health that is used to measure the worth of player strikes and many more add-onal features.

The two ranges, classic and beta come with single player and multiplayer option. In classic level, the player is expected to both build or destruct the structures in the game without being how to get your free minecraft keen about dodging and defending himself against the attackers and surviving other risks. Nevertheless, in this level, there are minimal numbers of blocks that the player can make use of in order to proceed with the constructions. In inclusion, the players have the choice to place or remove any block at the same time, no matter the differences. Apart from manipulation of these blocks, the gamers will not have any other environment for interactions in the virtual world. However, it is possible in beta level.

opposite, the beta level, which is still being updated and developed, has more features where the gamers are suppose to build structures using mining strategies, survive attackers and many more limiting blocks and obstacles. Apart from the building of structures, the cubes can also be employd to craft various items like chests, mine carts, weapons, buckets and many more. Also, there is more than one environmental interaction in this mode and the player might do cultivation, hunt animals or mine minerals. Other items such as player health and inventory are also present in this mode. Should you have any queries with regards to in which and also tips on how to use Minecraft Codes Generator, you'll be able to contact us in our web-site. While taking part in, yielding to monsters attacks and environmental risks may cause a reduction in value on the health meter, which can be regained by taking various things within the game.

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