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{{Technical|date=February 2010}}
{{Unreferenced|date=February 2010}}
}}
 
This table relates to the computational cost for certain operations used in [[elliptic curve cryptography]], used in practice for strong cryptographic security of a [[public key]] system. The columns of the table are labelled by various computational steps. The rows of the table are for different models of [[elliptic curve]]s.
The table below contains the time-cost for these operations:
 
==Abbreviations for the operations==
 
In the next section a table of all the costs of some of the possible operations in elliptic curves is given. In some applications of [[elliptic curve cryptography]] and the elliptic curve method of factorization ([[Lenstra elliptic curve factorization|ECM]]) it is necessary to consider the scalar multiplication [''n'']''P''. So, one way to do this is to compute successively:
 
: <math> P,\quad [2]P=P+P,\quad [3]P=[2]P+P,  \dots  , [n]P=[n-1]P+P </math>
 
But, it is faster to use [[Exponentiation by squaring|double-and-add method]], e.g. [5]''P'' = [2]([2]P) + ''P''.
In general to compute [''k'']''P'',write
 
<math> k=\sum_{i\le l}k_i2^i </math>
 
with ''k''<sub>''i''</sub> in {0,1} and <math>l=[log_2 k]</math>, ''k''<sub>''l'' = 1, then:
 
<math> [2](....([2]([2]([2]([2]([2]P+[k_{(l-1)}]P)+[k_{(l-2)}]P)+[k_{(l-3)}]P)+ \dots ) \dots +[k_1]P)+[k_0]P=
[2^l]P+[k_{(l-1)}2^{l-1}]P+ \dots  +[k_12]P+[k_0]P </math>.
 
Note that, this simple algorithm takes at most ''2l'' steps and each step consists of a doubling and (if ''k''<sub>''i''</sub> &ne; 0) adding two points. So, this is one of the reasons why addition and doubling formulas are defined.
Furthermore, this method is applicable to any group and if the group law is written multiplicatively, the double-and-add algorithm is instead called [[Exponentiation by squaring|square-and-multiply algorithm]].
 
For more information about other possible operations on elliptic curves see http://hyperelliptic.org/EFD/g1p/index.html.
 
To see what adding (ADD) and doubling (DBL) points on elliptic curves mean, see  [[elliptic curve#The group law|The group law]].
 
These are the operations considered in the table below:
<poem>
DBL - Doubling
ADD - Addition
mADD - Mixed addition: addition of an input that has been scaled to have ''Z''-coordinate 1.
mDBL - Mixed doubling: doubling of an input that has been scaled to have ''Z'' coordinate 1.
TPL - Tripling.
</poem>
 
==Tabulation==
 
Under different assumptions on the multiplication, addition, inversion for the elements in some fixed [[field (mathematics)|field]], the time-cost of these operations varies.
In this table it is assumed that:
 
: I = 100M, S = 1M, *param = 0M, add = 0M, *const = 0M
 
This means that 100 multiplications (M) are required to invert (I) an element; 1 multiplication is required to compute the square (S) of an element; no multiplication is needed to multiply an element by a parameter (*param), by a constant (*const), nor to add two elements.
 
For more information about other results obtained with different assumptions, see http://hyperelliptic.org/EFD/g1p/index.html
 
{| class="wikitable"
|-
! Curve shape, representation
! DBL
! ADD
!mADD
!mDBL
!TPL
|-
|align=center|[[Elliptic curve|Short Weierstrass projective]]
|11
|14
|11
|8
|
|-
|align=center|[[Elliptic curve|Short Weierstrass projective with a4=-1]]
|11
|14
|11
|8
|
|-
|align=center|[[Elliptic curve|Short Weierstrass projective with a4=-3]]
|10
|14
|11
|8
|
|-
|align=center|[[Tripling-oriented Doche–Icart–Kohel curve]]
|9
|17
|11
|6
|12
|-
|align=center|[[Hessian curve#Extended coordinates|Hessian curve extended]]
|9
|12
|11
|9
|
|-
|align=center|[[Hessian curves|Hessian curve projective]]
|8
|12
|10
|6
|14
|-
|align=center|[[Jacobian curve#Alternative coordinates for the Jacobi quartic|Jacobi quartic XYZ]]
|8
|13
|11
|5
|
|-
|align=center|[[Jacobian curve#Alternative coordinates for the Jacobi quartic|Jacobi quartic doubling-oriented XYZ]]
|8
|13
|11
|5
|
|-
|align=center|[[Twisted Hessian curves|Twisted Hessian curve projective]]
|8
|12
|12
|8
|14
|-
|align=center|[[Doubling-oriented Doche–Icart–Kohel curve]]
|7
|17
|12
|6
|
|-
|align=center|[[Jacobian curve|Jacobi intersection projective]]
|7
|14
|12
|6
|14
|-
|align=center|[[Jacobian curve#Extended coordinates|Jacobi intersection extended]]
|7
|12
|11
|7
|16
|-
|align=center|[[Twisted Edward Curves|Twisted Edwards projective]]
|7
|11
|10
|6
|
|-
|align=center|[[Twisted Edward Curves|Twisted Edwards Inverted]]
|7
|10
|9
|6
|
|-
|align=center|[[Twisted Edward Curves|Twisted Edwards Extended]]
|8
|9
|8
|7
|
|-
|align=center|[[Edwards curves|Edwards projective]]
|7
|11
|9
|6
|13
|-
|align=center|[[Jacobian curve#Alternative coordinates for the Jacobi quartic|Jacobi quartic doubling-oriented XXYZZ]]
|7
|11
|9
|6
|14
|-
|align=center|[[Jacobian curve#Alternative coordinates for the Jacobi quartic|Jacobi quartic XXYZZ]]
|7
|11
|9
|6
|14
|-
|align=center|[[Jacobian curve#Alternative coordinates for the Jacobi quartic|Jacobi quartic XXYZZR]]
|7
|10
|9
|7
|15
|-
|align=center|[[Edwards curve#Inverted Edwards coordinates|Edwards curve inverted]]
|7
|10
|9
|6
|
|-
|align=center|[[Montgomery curve]]
|4
|
|
|3
|
|}
 
==References==
 
*http://hyperelliptic.org/EFD/g1p/index.html
 
{{DEFAULTSORT:Table Of Costs Of Operations In Elliptic Curves}}
[[Category:Elliptic curve cryptography]]
[[Category:Finite fields]]
[[Category:Computational number theory]]
[[Category:Cryptographic attacks]]
[[Category:Elliptic curves]]

Latest revision as of 10:19, 8 February 2014

There are many types of bike brakes for different applications. Various means of transportation come second - from yachts or boats and cars to motorcycles. The trail then follows the side of the glacial Lake Pukaki through to Twizel, the town build in the 60s only to house workers from the Waitaki Hydroelectricity Scheme. If they aren't a know brand, don't panic, I'm certain they'll be fit for purpose, but bear in mind if they look really cheap then you may struggle with things such as adjustment of your gears. Two hunters may choose to use trailers if their distance from the truck is over four or five miles.

Similarly mountain bike wheels are also an important feature. Young children can hardly wait for their first Hot Wheels or tricycle. Fitting is logical, the instructions are clear few tools are required. - Get a bike that is lightweight and made out of carbon fiber materials. Browse a number of the on-line forums to determine what other riders like and do not like about their bicycles.

For those who have just about any queries about in which in addition to how to employ Here is your mountain bike sizing., you possibly can e mail us on our site. The "derailleur" attached to the gear shifter moves the chain from one ring to another. There are two main types of mountain and road bike stems. Please remember the stem lengths can impact just how reactive the bike is. The Grand Rapids area is becoming well known as a cycling destination. These are stable forks whose weights are directly in proportion to their durability.

Read product reviews and cycling magazines, research online, and ask for advice at your local bike shop. Mountain Bike shoes are available in many varieties and options. This article is to help you when buying a new bike. There is no shortage of videos of mountain bike crashes from around the world, with many riders with appropriate safety equipment escaping serious injury, while others were not that fortunate. Within 2 weeks of breathing exercises, the warm up period disappeared.

If you are just out there to like your journey or to get from position to put efficiently, an ebike electric mountain bike, electric mountain bike, electric bikes for sale is the great excess boost to your experience that will permit you go farther, more time and have much more electrical power above the course of your ride, letting you to appreciate the journey a little extra. There is anything from shocks to gears, special wheels, exclusive take care of bars that can be immediately switched to match the terrain you're riding on and so substantially much more. One of the main concerns for any cyclist is safety, having the right kit is a must. Whether it's a testament to South Dakota's commitment to recycling, or just whimsy, Bone Collector embodies the features of the best bike trails in the world: Sedona, Moab, Flagstaff, Whistler and Tahoe, without the long drive or expensive plane ticket. From there you can connect to the west switchback trail simply call "Switchbacks" or "The Old Switchbacks".

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