|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{multiple issues|
| | Leverage gives us the option to control a lot of money with just a small percentage of that money being ours. When trading stocks, if you want to buy $1,000 of stock, you must invest at least $500. This gives you a [http://Pinterest.com/search/pins/?q=leverage leverage] of 2:1.<br><br>There is lots of people that make tremendous cash day trading stocks. They will buy and sell stocks throughout the day and it's not uncommon for them to execute dozens, even hundreds of trades every day. However, keep in mind that these individuals do this as their fulltime job. They spend their workday in front of their computers, continuously analyzing the markets and new info that comes out.<br><br>The traditional way of earning money would be to get a job and to work based on your expertise or trained skill. Your services or the products you make are rewarded with a salary or a profit for the products sold. However, with the emergence of financial systems, more and more ways of earning money - without doing any real work - have been created. This is through investing in stocks and other markets. One such popular market is the Free Forex trading system.<br><br>After you have bought some shares from a company, you will be issued with a share certificate that will serve as a proof that you own the shares. In the certificate, you will find the nominal or par, value of shares at the time the certificate was issued. It does not have any relevance to the current market value of your shares but it is required to be http://news.goldgrey.org/gold-futures/ - [http://news.goldgrey.org/gold-futures/ Read Homepage] - printed on the share certificate for legal purposes.<br><br>If you choose to join in penny stock trading, try to find a good broker. Brokers are people who offer you broker accounts for nominal fees. Once you open the account, you can start investing in stocks and the broker will deduct the small fee from your account. The fee is issued to pay for the brokers expenses. Never expect to get any advice from brokers because these people are account managers and not advisors.<br><br>Become familiar with the futures. On the NASDAQ, [http://Www.google.co.uk/search?hl=en&gl=us&tbm=nws&q=futures&gs_l=news futures] do play an important role as the stocks usually move upward or downward with the futures. For example, you should never short a stock if the futures are in a strong upward trend.<br><br>Overall, day trading is a very rewarding career for you to go into. While it is risky, you will reap the rewards that come with working from home. YOu can trade whenever the market is on and you can finally start realizing some of the return that you deserve. |
| {{Refimprove|date=March 2011}}
| |
| {{Orphan|date=February 2009}}
| |
| }}
| |
| | |
| In 1998 [[Gerhard Frey]] firstly proposed using '''trace zero varieties''' for cryptographic purpose. These varieties are subgroups of the divisor class group on a low genus hyperelliptic curve defined over a [[finite field]]. These groups can be used to establish [[Public-key cryptography|asymmetric cryptography]] using the [[discrete logarithm]] problem as cryptographic primitive.
| |
| | |
| Trace zero varieties feature a better scalar multiplication performance than elliptic curves. This allows a fast arithmetic in this groups, which can speed up the calculations with a factor 3 compared with elliptic curves and hence speed up the cryptosystem.
| |
| | |
| Another advantage is that for a groups of cryptographically relevant size, the order of the group can simply be calculated using the characteristic polynomial of the Frobenius endomorphism. This is not the case, for example, in [[elliptic curve cryptography]] when the group of points of an elliptic curve over a prime field is used for cryptographic purpose.
| |
| | |
| However to represent an element of the trace zero variety more bits are needed compared with elements of elliptic or hyperelliptic curves. Another disadvantage, is the fact, that it is possible to reduce the security of the TZV of <sup>1</sup>/<sub>6</sub><sup>th</sup> of the bit length using cover attack.
| |
| | |
| == Mathematical background ==
| |
| A [[hyperelliptic curve]] ''C'' of genus ''g'' over a prime field <math>\mathbb{F}_q</math> where ''q'' = ''p''<sup>''n''</sup> (''p'' prime) of odd characteristic is defined as
| |
| | |
| : <math>
| |
| C:~y^2 + h(x)y = f(x),
| |
| </math>
| |
| | |
| where ''f'' monic, deg(''f'') = 2''g'' + 1 and deg(''h'') ≤ g. The curve has at least one <math>\mathbb{F}_q</math>-rational Weierstraßpoint.
| |
| | |
| The [[Jacobian variety]] <math>J_C(\mathbb{F}_{q^n})</math> of ''C'' is for all finite extension <math>\mathbb{F}_{q^n}</math> isomorphic to the ideal class group <math>\operatorname{Cl}(C/\mathbb{F}_{q^n})</math>. With the ''Mumford's representation'' it is possible to represent the elements of <math>J_C(\mathbb{F}_{q^n})</math> with a pair of polynomials ''[u, v]'', where ''u'', ''v'' ∈ <math>\mathbb{F}_{q^n}[x]</math>.
| |
| | |
| The ''Frobenius endomorphism'' σ is used on an element ''[u, v]'' of <math>J_C(\mathbb{F}_{q^n})</math> to raise the power of each coefficient of that element to ''q'': σ(''[u, v]'') = [''u''<sup>q</sup>(x), v<sup>q</sup>(x)]. The characteristic polynomial of this endomorphism has the following form:
| |
| | |
| : <math>
| |
| \chi(T) = T^{2g} + a_1T^{2g-1} + \cdots + a_gT^g + \cdots + a_1q^{g-1}T + q^g,
| |
| </math>
| |
| where a<sub>i</sub> in {{Unicode|ℤ}}
| |
| | |
| With the ''Hasse–Weil theorem'' it is possible to receive the group order of any extension field <math>\mathbb{F}_{q^n}</math> by using the complex roots τ<sub>i</sub> of χ(''T''):
| |
| | |
| : <math>
| |
| |J_C(\mathbb{F}_{q^n})| = \prod_{i=1}^{2g} (1 - \tau_i^n)
| |
| </math>
| |
| | |
| Let ''D'' be an element of the <math>J_C(\mathbb{F}_{q^n})</math> of ''C'', then it is possible to define an endomorphism of <math>J_C(\mathbb{F}_{q^n})</math>, the so-called ''trace of D'':
| |
| | |
| : <math>
| |
| \operatorname{Tr}(D) = \sum_{i=0}^{n-1} \sigma^i(D) = D + \sigma(D) + \cdots + \sigma^{n-1}(D)
| |
| </math>
| |
| | |
| Based on this endomorphism one can reduce the Jacobian variety to a subgroup ''G'' with the property, that every element is of trace zero:
| |
| | |
| : <math>
| |
| G = \{ D \in J_C(\mathbb{F}_{q^n})~|~\text{Tr}(D) = \textbf{\textit{0}} \}, ~~~(\textbf{\textit{0}} \text{ neutral element in } J_C(\mathbb{F}_{q^n})
| |
| </math>
| |
| | |
| ''G'' is the kernel of the trace endomorphism and thus ''G'' is a group, the so-called '''trace zero (sub)variety''' (TZV) of <math>J_C(\mathbb{F}_{q^n})</math>.
| |
| | |
| The intersection of ''G'' and <math>J_C(\mathbb{F}_{q})</math> is produced by the ''n''-torsion elements of <math>J_C(\mathbb{F}_{q})</math>. If the greatest common divisor <math>\gcd(n, |J_C(\mathbb{F}_q)|) = 1</math> the intersection is empty and one can compute the group order of ''G'':
| |
| | |
| : <math>
| |
| |G| = \dfrac{|J_C(\mathbb{F}_{q^n})|}{|J_C(\mathbb{F}_q)|} = \dfrac{\prod_{i=1}^{2g} (1 - \tau_i^n)}{ \prod_{i=1}^{2g} (1 - \tau_i)}
| |
| </math>
| |
| | |
| The actual group used in cryptographic applications is a subgroup ''G<sub>0</sub>'' of ''G'' of a large prime order ''l''. This group may be ''G'' itself.<ref>G. Frey and T. Lange: "Mathematical background of public key cryptography"</ref><ref>T. Lange: "Trace zero subvariety for cryptosystems"</ref>
| |
| | |
| There exist three different cases of cryptograpghical relevance for TZV:<ref>R. M. Avanzi and E. Cesena: "Trace zero varieties over fields of characteristic 2 for cryptographic applications"</ref>
| |
| *''g'' = 1, ''n'' = 3
| |
| *''g'' = 1, ''n'' = 5
| |
| *''g'' = 2, ''n'' = 3
| |
| | |
| == Arithmetic ==
| |
| | |
| The arithmetic used in the TZV group ''G<sub>0</sub>'' based on the arithmetic for the whole group <math>J_C(\mathbb{F}_{q^n})</math>, But it is possible to use the ''Frobenius endomorphism'' σ to speed up the scalar multiplication. This can be archived if ''G<sub>0</sub>'' is generated by ''D'' of order ''l'' then ''σ(D) = sD'', for some integers ''s''. For the given cases of TZV ''s'' can be computed as follows, where ''a''<sub>i</sub> come from the characteristic polynomial of the Frobenius endomorphism :
| |
| | |
| * For ''g'' = 1, ''n'' = 3: <math>s = \dfrac {q-1} {1 - a_1} \bmod{\ell} </math>
| |
| | |
| * For ''g'' = 1, ''n'' = 5: <math>s = \dfrac {q^2-q-a_1^2q+a_1q+1} {q-2a_1q+a_1^3-a_1^2+a_1-1} \bmod{\ell} </math>
| |
| | |
| * For ''g'' = 2, ''n'' = 3: <math>s = - \dfrac {q^2-a_2+a_1} {a_1q-a_2+1} \bmod{\ell}</math>
| |
| | |
| Knowing this, it is possible to replace any scalar multiplication ''mD (|m| ≤ l/2)'' with:
| |
| | |
| : <math>
| |
| m_0D + m_1\sigma(D) + \cdots + m_{n-1}\sigma^{n-1}(D), ~~~~\text{where }m_i = O(\ell^{1/(n-1)}) = O(q^g)
| |
| </math>
| |
| | |
| With this trick the multiple scalar product can be reduced to about 1/(''n'' − 1)<sup>th</sub> of doublings necessary for calculating ''mD'', if the implied constants are small enough.<ref>R. M. Avanzi and E. Cesena: "Trace zero varieties over fields of characteristic 2 for cryptographic applications"</ref><ref>T. Lange: "Trace zero subvariety for cryptosystems"</ref>
| |
| | |
| == Security ==
| |
| The security of cryptographic systems based on trace zero subvarieties according of the results of the papers<ref>T. Lange: "Trace zero subvariety for cryptosystems"</ref><ref>R. M. Avanzi and E. Cesena: "Trace zero varieties over fields of characteristic 2 for cryptographic applications"</ref>
| |
| comparable to the security of hyper-elliptic curves of low genus ''g' '' over <math>\mathbb{F}_{p'}</math>, where ''p' '' ~ (''n'' − 1)(''g/g' '') for ''|G|'' ~128 bits.
| |
| | |
| For the cases where ''n'' = 3, ''g'' = 2 and ''n'' = 5, ''g'' = 1 it is possible to reduce the security for at most 6 bits, where ''|G|'' ~ 2<sup>256</sup>, because one can not be sure that ''G'' is contained in a Jacobian of a curve of genus 6. The security of curves of genus 4 for similar fields are far less secure. | |
| | |
| == Cover attack on a trace zero crypto-system ==
| |
| The attack published in<ref>C. Diem and J. Scholten: "An attack on a trace-zero cryptosystem"</ref>
| |
| shows, that the DLP in trace zero groups of genus 2 over finite fields of characteristic diverse than 2 or 3 and a field extension of degree 3 can be transformed into a DLP in a class group of degree 0 with genus of at most 6 over the base field. In this new class group the DLP can be attacked with the index calculus methods. This leads to a reduction of the bit length <sup>1</sup>/<sub>6</sub><sup>th</sup>.
| |
| | |
| == Notes ==
| |
| {{reflist|2}}
| |
| | |
| == References ==
| |
| * G. Frey and T. Lange: "Mathematical background of public key cryptography", Technical report, 2005{{Refimprove-inline|date=March 2011}}
| |
| * R. M. Avanzi and E. Cesena: "Trace zero varieties over fields of characteristic 2 for cryptographic applications", Technical report, 2007{{Refimprove-inline|date=March 2011}}
| |
| * T. Lange: "Trace zero subvariety for cryptosystems", Technical report, 2003, http://eprint.iacr.org/2003/094, 2003 {{Refimprove-inline|date=March 2011}}
| |
| * C. Diem and J. Scholten: "An attack on a trace-zero cryptosystem"{{Refimprove-inline|date=March 2011}}
| |
| * M. Wienecke: "Cryptography on Trace-Zero Varieties", ITS-Seminar paper, http://www.crypto.rub.de/its_seminar_ws0708.html, 2008
| |
| * A. V. Sutherland: "101 useful trace zero varieties", http://www-math.mit.edu/~drew/TraceZeroVarieties.html, 2007
| |
| | |
| [[Category:Cryptography]]
| |
Leverage gives us the option to control a lot of money with just a small percentage of that money being ours. When trading stocks, if you want to buy $1,000 of stock, you must invest at least $500. This gives you a leverage of 2:1.
There is lots of people that make tremendous cash day trading stocks. They will buy and sell stocks throughout the day and it's not uncommon for them to execute dozens, even hundreds of trades every day. However, keep in mind that these individuals do this as their fulltime job. They spend their workday in front of their computers, continuously analyzing the markets and new info that comes out.
The traditional way of earning money would be to get a job and to work based on your expertise or trained skill. Your services or the products you make are rewarded with a salary or a profit for the products sold. However, with the emergence of financial systems, more and more ways of earning money - without doing any real work - have been created. This is through investing in stocks and other markets. One such popular market is the Free Forex trading system.
After you have bought some shares from a company, you will be issued with a share certificate that will serve as a proof that you own the shares. In the certificate, you will find the nominal or par, value of shares at the time the certificate was issued. It does not have any relevance to the current market value of your shares but it is required to be http://news.goldgrey.org/gold-futures/ - Read Homepage - printed on the share certificate for legal purposes.
If you choose to join in penny stock trading, try to find a good broker. Brokers are people who offer you broker accounts for nominal fees. Once you open the account, you can start investing in stocks and the broker will deduct the small fee from your account. The fee is issued to pay for the brokers expenses. Never expect to get any advice from brokers because these people are account managers and not advisors.
Become familiar with the futures. On the NASDAQ, futures do play an important role as the stocks usually move upward or downward with the futures. For example, you should never short a stock if the futures are in a strong upward trend.
Overall, day trading is a very rewarding career for you to go into. While it is risky, you will reap the rewards that come with working from home. YOu can trade whenever the market is on and you can finally start realizing some of the return that you deserve.