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In [[mathematics]], the '''Twisted Hessian curve''' represents a generalization of [[Hessian curve]]s; it was introduced in [[elliptic curve cryptography]] to speed up the addition and doubling formulas and to have strongly unified arithmetic. In some operations (see the last sections), it is close in speed to [[Edwards curve]]s.
While this mindset is certainly understandable, it simply isn't practical at that price point. Various means of transportation come second - from yachts or boats and cars to motorcycles. The downhill bikes have both front and rear suspension, disc breaks and are very strong. If the skin diameter is smaller than a quarter do not be surprised if you must change them a couple of instances per season to keep the bike working reliably. There are many times you will need to grip the handle of your bike tightly when you negotiate difficult trails. <br><br>If you want to get information about some really special and unique forks visit my site. Young children can hardly wait for their first Hot Wheels or tricycle. You can also purchase helmets and other safety equipment, lights, locks, saddles, racks, packs, trainers, bike maintenance gear, trailers, team wear, Trek souvenirs, and even computers for your bike. Just to give you an idea, mine has three settings, those being; *Completely off, so that I am riding it as a normal bike; *Power assist, where the power kicks in after a couple of revolutions of the pedals and then remains on as long as the pedals are turning, (there is a high and low setting in this mode). Then sense the direct route it will take to go from your nose to the target location and then back out through the nose. <br><br>Mountain bikes in Arizona have two great rides in Sedona, Highline and the Dry Creek Loop. If you have any thoughts pertaining to the place and how to use [http://igrice.square7.net/profile/fawolfe Transfering to mountain bike sizing.], you can call us at the web site. There are two main types of mountain and road bike stems. The whole set-up, including your bag, should weigh no more than 20kgs (44lbs). The Grand Rapids area is becoming well known as a cycling destination. If you do, below is a list of items that you just would possibly wish to look at prior to shopping for a new mountain bike. <br><br>The tracks are mostly dirt paths on outdoor scenery. As a form of transport in itself bike riding is great but because of the design of these bikes you can travel over harsher terrain which will lead to you tossing in your old bike and traveling to work on your newly discovered fun machine. As the tracks are very rough it is far more difficult on a hardtail bike. The internet will do just fine, maybe even better, as there are countless of online stores and biking websites from which you may get the information you need. 7,934,739) CVA is an award-winning cancellation style and now it is recognized with a US Apparent. <br><br>In the recent article section, you can find the list of biking related articles with a detailed description present on the official website. The bike should be examined to check it is working correctly. One of the main concerns for any cyclist is safety, having the right kit is a must. Sign up to get a mountain bike expedition and relive people childhood biking wonder and thrills with a whole new tier. Which leads me to my favourite materials for hardtails.
 
==Definition==
[[Image:Twisted Hessian curve.svg|300px|right|thumb|A Twisted Hessian curve of equation <math>10x^3+y^3+1=15xy</math>]]
Let ''K'' be a [[field (mathematics)|field]]. According to<ref>{{cite web|url=http://hyperelliptic.org/EFD/g1p/auto-twistedhessian.html|title=Twisted Hessian curves|accessdate=28 February 2010}}</ref> twisted Hessian curves were introduced by [[Daniel J. Bernstein|Bernstein]], Lange,
and Kohel.
 
The twisted Hessian form in ''[[affine space|affine coordinates]]'' is given by:
 
<math> a\cdot x^3+y^3+1=d\cdot x\cdot y </math>
 
and in ''[[projective space|projective coordinates]]'':
 
<math> a\cdot X^3+Y^3+Z^3=d\cdot X\cdot Y\cdot Z </math>
 
where <math>x=\frac{X}{Z}</math>  and  <math> y=\frac{Y}{Z}</math>  and  ''a'', ''d'' in ''K''
 
Note that these curves are [[birational geometry|birationally equivalent]] to [[Hessian curves]].
 
The Hessian curve is just a special case of Twisted Hessian curve, with a=1.
 
Considering the equation ''a'' &middot; ''x''<sup>3</sup> + ''y''<sup>3</sup> + 1 = ''d'' &middot; ''x'' &middot; ''y'', note that:
 
if ''a'' has a cube root in ''K'', there exists a unique ''b'' such that ''a'' = ''b''<sup>3</sup>.Otherwise, it is necessary to consider an [[field extension|extension field]] of ''K'' (e.g., ''K''(''a''<sup>1/3</sup>)). Then, since ''b''<sup>3</sup> &middot; ''x''<sup>3</sup> = ''bx''<sup>3</sup>, defining ''t'' = ''b'' &middot; ''x'', the following equation is needed (in Hessian form) to do the transformation:
 
<math> t^3+y^3+1=d\cdot x\cdot y </math>.
 
This means that Twisted Hessian curves are birationally equivalent to elliptic curve in [[Elliptic curve|Weierstrass form]].
 
==Group law==
 
It is interesting to analyze the [[elliptic curve#The group law|group law]] of the elliptic curve, defining the addition and doubling formulas (because the [[Power_analysis#Simple_power_analysis|simple power analysis]] and [[Power_analysis#Differential_power_analysis|differential power analysis]] attacks are based on the running time of these operations). In general, the group law is defined in the following way: if three points lies in the same line then they sum up to zero. So, by this property, the [[explicit formula]]s{{dn|date=December 2013}} for the group law depend on the curve shape.
 
Let ''P'' = (''x''<sub>1</sub>, ''y''<sub>1</sub>) be a point, then its inverse is &minus;''P'' = (''x''<sub>1</sub>/''y''<sub>1</sub>, 1/''y''<sub>1</sub>) in the plane.
In projective coordinates, let ''P'' = (''X'' : ''Y'' : ''Z'') be one point, then &minus;''P'' = (''X''<sub>1</sub>/''Y''<sub>1</sub> : 1/''Y''<sub>1</sub> : ''Z'')<!-- ?? --> is the inverse of P.
 
Furthermore, the [[neutral element]] (in affine plane) is: &theta; = (0, &minus;1) and in projective coordinates: &theta; = (0 : &minus;1 : 1).
 
In some applications of [[elliptic curve cryptography]] and the elliptic curve method of [[integer factorization]] ([[Lenstra elliptic curve factorization|ECM]]) it is necessary to compute the [[scalar multiplication]]s of ''P'', say ''[n]P'' for some [[integer]] ''n'', and they are based on the [[Exponentiation by squaring|double-and-add]] method; so the addition and doubling formulas are needed.
 
The addition and doubling formulas for this [[elliptic curve]] can be defined, using the affine coordinates to simplify the notation:
 
===Addition formulas===
 
Let ''p'' = (''x''<sub>1</sub>, ''y''<sub>1</sub>) and ''Q'' = (''x''<sub>2</sub>, ''y''<sub>2</sub>); then, ''R'' = ''P'' + ''Q'' = (''x''<sub>3</sub>, ''y''<sub>3</sub>) is given by the following equations:
 
<math>x_3=\frac{x_1-y_1^3\cdot x_1}{a\cdot y_1\cdot x_1^3-y_1} </math>
 
<math> y_3=\frac{y_1^3-a\cdot x_1^3}{a\cdot y_1\cdot x_1^3-y_1} </math>
 
===Doubling formulas===
 
Let ''P'' = (''x'', ''y''); then [2]''P'' = (''x''<sub>1</sub>, ''y''<sub>1</sub>) is given by the following equations:
 
<math> x_1=\frac{x-y^3\cdot x}{a\cdot y\cdot x^3-y} </math>
 
<math> y_1=\frac{y^3-a\cdot x^3}{a\cdot y\cdot x^3-y} </math>
 
==Algorithms and examples==
 
Here some efficient algorithms of the addition and doubling law are given; they can be important in cryptographic computations, and the projective coordinates are used to this purpose.
 
===Addition===
 
<math> A = X_1\cdot Z_2 </math>
 
<math>B = Z_1\cdot Z_2 </math>
 
<math>C = Y_1X_2 </math>
 
<math>D = Y_1\cdot Y_2 </math>
 
<math>E = Z_1\cdot Y_2 </math>
 
<math>F = a\cdot X_1\cdot X_2 </math>
 
<math>X_3 = A\cdot B-C\cdot D </math>
 
<math>Y_3 = D\cdot E-F\cdot A </math>
 
<math>Z_3 = F\cdot C-B\cdot E </math>
 
The cost of this algorithm is 12 multiplications, one multiplication by a (constant) and 3 additions.
 
'''Example:'''
 
let ''P''<sub>1</sub> = (1 : &minus;1 : 1) and ''P''<sub>2</sub> = (&minus;2 : 1 : 1) be points over a twisted Hessian curve with a=2 and d=-2.Then ''R'' = ''P''<sub>1</sub> + ''P''<sub>2</sub> is given by:
 
<math> A=-1; B=-1; C=-1; D=-1; E=1; F=2; </math>
:<math>x_3=0</math>
:<math>y_3=-3</math>
:<math>z_3=-3</math>
 
That is, ''R''= (0 : &minus;3 : &minus;3).
 
===Doubling===
 
<math> D = X_1^3 </math>
 
<math> E = Y_1^3 </math>
 
<math> F = Z_1^3 </math>
 
<math> G = a\cdot D </math>
 
<math> X_3 = X_1\cdot (E-F) </math>
 
<math> Y_3 = Z_1\cdot (G-E) </math>
 
<math> Z_3 = Y_1\cdot (F-G) </math>
 
The cost of this algorithm is 3 multiplications, one multiplication by constant, 3 additions and 3 cube powers.
This is the best result obtained for this curve.
 
'''Example:'''
 
let ''P'' = (1 : &minus;1 : 1) be a point over the curve defined by a=2 and d=-2 as above, then ''R'' = [2]''P'' = (''x''<sub>3</sub> : ''y''<sub>3</sub> : ''z''<sub>3</sub>) is given by:
 
<math> D=1; E=-1; F=1; G=-4; </math>
 
:<math>x_3=-2</math>
:<math>y_3=-3</math>
:<math>z_3=-5</math>
 
That is ''R'' = (&minus;2 : &minus;3 : 5).
 
==See also==
* [[Table of costs of operations in elliptic curves]]
 
==External links==
* http://hyperelliptic.org/EFD/g1p/index.html
 
==References==
 
<references/>
* http://hyperelliptic.org/EFD/g1p/auto-twistedhessian.html
 
{{DEFAULTSORT:Twisted Hessian Curves}}
[[Category:Elliptic curves]]
[[Category:Elliptic curve cryptography]]

Latest revision as of 07:04, 24 July 2014

While this mindset is certainly understandable, it simply isn't practical at that price point. Various means of transportation come second - from yachts or boats and cars to motorcycles. The downhill bikes have both front and rear suspension, disc breaks and are very strong. If the skin diameter is smaller than a quarter do not be surprised if you must change them a couple of instances per season to keep the bike working reliably. There are many times you will need to grip the handle of your bike tightly when you negotiate difficult trails.

If you want to get information about some really special and unique forks visit my site. Young children can hardly wait for their first Hot Wheels or tricycle. You can also purchase helmets and other safety equipment, lights, locks, saddles, racks, packs, trainers, bike maintenance gear, trailers, team wear, Trek souvenirs, and even computers for your bike. Just to give you an idea, mine has three settings, those being; *Completely off, so that I am riding it as a normal bike; *Power assist, where the power kicks in after a couple of revolutions of the pedals and then remains on as long as the pedals are turning, (there is a high and low setting in this mode). Then sense the direct route it will take to go from your nose to the target location and then back out through the nose.

Mountain bikes in Arizona have two great rides in Sedona, Highline and the Dry Creek Loop. If you have any thoughts pertaining to the place and how to use Transfering to mountain bike sizing., you can call us at the web site. There are two main types of mountain and road bike stems. The whole set-up, including your bag, should weigh no more than 20kgs (44lbs). The Grand Rapids area is becoming well known as a cycling destination. If you do, below is a list of items that you just would possibly wish to look at prior to shopping for a new mountain bike.

The tracks are mostly dirt paths on outdoor scenery. As a form of transport in itself bike riding is great but because of the design of these bikes you can travel over harsher terrain which will lead to you tossing in your old bike and traveling to work on your newly discovered fun machine. As the tracks are very rough it is far more difficult on a hardtail bike. The internet will do just fine, maybe even better, as there are countless of online stores and biking websites from which you may get the information you need. 7,934,739) CVA is an award-winning cancellation style and now it is recognized with a US Apparent.

In the recent article section, you can find the list of biking related articles with a detailed description present on the official website. The bike should be examined to check it is working correctly. One of the main concerns for any cyclist is safety, having the right kit is a must. Sign up to get a mountain bike expedition and relive people childhood biking wonder and thrills with a whole new tier. Which leads me to my favourite materials for hardtails.

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