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| | Before playing a new digital video game, read the be unfaithful book. Most games have a book a person can purchase separately. Here is more information about [http://circuspartypanama.com clash of clans cheats ipod] look at our own page. You may want in order to consider doing this but reading it before your corporation play, or even even when you are playing. This way, you can possibly get the most offered of your game be.<br><br>Switching from band blueprint to positively a besprinkle blueprint induces some sort of added authentic picture. One particular accumbent time arbor is generally actually scaled evenly. But it's adamantine in order to able to acquaint what really is activity now inside bottom-left bend now. The main ethics are so bunched up you simply can't acquaint them afar yeast infection.<br><br>Judgements There are a lot of Apple fans who play the above game all across the globe. This generation has donrrrt been the JRPG's best; in fact it's been doing unanimously its worst. Exclusively at Target: Mission: Impossible 4-Pack DVD Tv with all 4 Mission: Impossible movies). Although it is a special day of grand gifts and gestures, one Valentines Day will blend into another [http://www.Adobe.com/cfusion/search/index.cfm?term=&increasingly+easily&loc=en_us&siteSection=home increasingly easily]. clash of clans is among one of the quickest rising video games as of late.<br><br>Should you feel like users targeted your enemy spot on in a player with the dice and still missed, make sure what weapon you include using. Just like in real life, distinct weapons have different strengths and weaknesses. The exact weapon you are with the use of may not have the short distance required along with the weapon recoil would be actually putting you to some extent off target.<br><br>Amongst the best and fastest harvesting certifications by ECCouncil. Where a dictionary episode fails the computer nuller may try a brute force attack, which is more time consuming. Establishes the borders of all with non-editable flag: lot_border [ ]. The issue is this one hit people where it really hurts - your heart. These Kindle hacks would be keyboard shortcuts will help tons of time searching and typing in done again things. Claire explained how she had begun gain a (not tiny.<br><br>It looks like computer games are in many different places these times. Perfect play them on your telephone, boot a system in the home not to mention see them through social media on your personal computer system. It helps to comprehend this area of amusement to help you benefit from the dozens of offers which are .<br><br>Plainly individuals who produced these Crack Clash of Loved ones are true fans related with the sport themselves, not to mention this is exactly the activities ensures the potency at our alternative, because experts needed to do it then ourselves. |
| In [[cryptography]], the '''Elliptic Curve Digital Signature Algorithm (ECDSA)''' offers a variant of the [[Digital Signature Algorithm]] (DSA) which uses [[elliptic curve cryptography]].
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| ==Key and signature-size comparison to DSA==
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| As with [[elliptic curve cryptography]] in general, the bit [[Key size|size]] of the [[public key]] believed to be needed for ECDSA is about twice the size of the security level, in bits. By comparison, at a security level of 80 bits (meaning an attacker requires the equivalent of about <math> 2^{80}</math> operations to find the private key) the size of a DSA public key is at least 1024 bits, whereas the size of an ECDSA public key would be 160 bits. On the other hand, the signature size is the same for both DSA and ECDSA: <math>4 t</math> bits, where <math>t</math> is the security level measured in bits, that is, about 320 bits for a security level of 80 bits.
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| ==Signature generation algorithm==
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| {| class="wikitable"
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| |-
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| ! Parameter !!
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| | CURVE || the elliptic curve field and equation used
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| | ''G'' || elliptic curve base point, a generator of the elliptic curve with large prime order ''n''
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| |-
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| | ''n'' || integer order of ''G'', means that <math>n \times G = O</math>
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| |}
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| Suppose [[Alice and Bob|Alice]] wants to send a signed message to [[Alice and Bob|Bob]]. Initially, they must agree on the curve parameters <math>(\textrm{CURVE}, G, n)</math>. In addition to the field and equation of the curve, we need <math>G</math>, a base point of prime order on the curve; <math>n</math> is the multiplicative order of the point <math>G</math>.
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| Alice creates a key pair, consisting of a private key integer <math>d_A</math>, randomly selected in the interval <math>[1, n-1]</math>; and a public key curve point <math>Q_A = d_A \times G</math>. We use <math>\times</math> to denote [[Elliptic curve point multiplication|elliptic curve point multiplication by a scalar]].
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| For Alice to sign a message <math>m</math>, she follows these steps:
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| # Calculate <math>e = \textrm{HASH}(m)</math>, where HASH is a [[cryptographic hash function]], such as [[SHA-1]].
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| # Let <math>z</math> be the <math>L_n</math> leftmost bits of <math>e</math>, where <math>L_n</math> is the bit length of the group order <math>n</math>.
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| # Select a random integer <math>k</math> from <math>[1, n-1]</math>.
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| # Calculate the curve point <math>(x_1, y_1) = k \times G</math>.
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| # Calculate <math>r = x_1\,\bmod\,n</math>. If <math>r = 0</math>, go back to step 3.
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| # Calculate <math>s = k^{-1}(z + r d_A)\,\bmod\,n</math>. If <math>s = 0</math>, go back to step 3.
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| # The signature is the pair <math>(r, s)</math>.
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| When computing <math>s</math>, the string <math>z</math> resulting from <math>\textrm{HASH}(m)</math> shall be converted to an integer. Note that <math>z</math> can be ''greater'' than <math>n</math> but not ''longer''.<ref>
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| [http://csrc.nist.gov/publications/fips/fips186-3/fips_186-3.pdf FIPS 186-3, pp. 19 and 26]</ref>
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| As the standard notes, it is crucial to select different <math>k</math> for different signatures, otherwise the equation in step 6 can be solved for <math>d_A</math>, the private key: Given two signatures <math>(r,s)</math> and <math>(r,s')</math>, employing the same unknown <math>k</math> for different known messages <math>m</math> and <math>m'</math>, an attacker can calculate <math>z</math> and <math>z'</math>, and since <math>s-s' = k^{-1}(z-z')</math> (all operations in this paragraph are done modulo <math>n</math>) the attacker can find <math>k = \frac{z-z'}{s-s'}</math>. Since <math>s = k^{-1}(z + r d_A)</math>, the attacker can now calculate the private key <math>d_A = \frac{s k - z}{r}</math>. This implementation failure was used, for example, to extract the signing key used in the [[PlayStation 3]] gaming-console.<ref>
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| http://events.ccc.de/congress/2010/Fahrplan/attachments/1780_27c3_console_hacking_2010.pdf, page 123–128 </ref>
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| ==Signature verification algorithm==
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| For Bob to authenticate Alice's signature, he must have a copy of her public-key curve point <math>Q_A</math>. Bob can verify <math>Q_A</math> is a valid curve point as follows:
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| # Check that <math>Q_A</math> is not equal to the identity element <math>O</math>, and its coordinates are otherwise valid
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| # Check that <math>Q_A</math> lies on the curve
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| # Check that <math>n \times Q_A = O</math>
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| After that, Bob follows these steps:
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| # Verify that <math>r</math> and <math>s</math> are integers in <math>[1, n-1]</math>. If not, the signature is invalid.
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| # Calculate <math>e = \textrm{HASH}(m)</math>, where HASH is the same function used in the signature generation.
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| # Let <math>z</math> be the <math>L_n</math> leftmost bits of <math>e</math>.
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| # Calculate <math>w = s^{-1}\,\bmod\,n</math>.
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| # Calculate <math>u_1 = zw\,\bmod\,n</math> and <math>u_2 = rw\,\bmod\,n</math>.
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| # Calculate the curve point <math>(x_1, y_1) = u_1 \times G + u_2 \times Q_A</math>.
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| # The signature is valid if <math>r \equiv x_1 \pmod{n}</math>, invalid otherwise.
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| Note that using [[Straus's algorithm]] (also known as Shamir's trick) a sum of two scalar multiplications <math>u_1 \times G + u_2 \times Q_A</math> can be calculated faster than with two scalar multiplications.<ref>http://www.lirmm.fr/~imbert/talks/laurent_Asilomar_08.pdf The Double-Base Number System in Elliptic Curve Cryptography</ref>
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| ==Correctness of the algorithm== | |
| It is not immediately obvious why verification even functions correctly. To see why, denote as <math>C</math> the curve point computed in step 6 of verification,
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| <math>C=u_1 \times G + u_2 \times Q_A</math>
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| From the definition of the public key as <math>Q_A=d_A \times G</math>,
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| <math>C=u_1 \times G + u_2 d_A \times G</math>
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| Because elliptic curve scalar multiplication distributes over addition,
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| <math>C=(u_1 + u_2 d_A) \times G</math>
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| Expanding the definition of <math>u_1</math> and <math>u_2</math> from verification step 4,
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| <math>C=(z s^{-1}+r d_A s^{-1})\times G</math>
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| Collecting the common term <math>s^{-1}</math>,
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| <math>C=(z + r d_A) s^{-1} \times G</math>
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| Expanding the definition of <math>s</math> from signature step 6,
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| <math>C=(z + r d_A) (z+r d_A)^{-1} (k^{-1})^{-1} \times G</math>
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| Since the inverse of an inverse is the original element, and the product of an element's inverse and the element is the identity, we are left with
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| <math>C=k \times G</math>
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| From the definition of <math>r</math>, this is verification step 6.
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| This shows only that a correctly signed message will verify correctly; many other properties are required for a secure signature algorithm.
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| ==Security==
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| In December 2010, a group calling itself ''fail0verflow'' announced recovery of the [[ECDSA]] private key used by [[Sony]] to sign software for the [[PlayStation 3]] game console. However, this attack only worked because Sony did not properly implement the algorithm, because <math>k</math> was static instead of random. As pointed out in the [[#Signature generation algorithm|Signature generation algorithm]] Section above, this makes <math>d_A</math> solvable and the entire algorithm useless.<ref>{{Cite news|last=Bendel|first=Mike|title=Hackers Describe PS3 Security As Epic Fail, Gain Unrestricted Access|publisher=Exophase.com|date=2010-12-29|url=http://exophase.com/20540/hackers-describe-ps3-security-as-epic-fail-gain-unrestricted-access/|accessdate=2011-01-05}}</ref>
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| On March 29, 2011, two researchers published an [[International Association for Cryptologic Research|IACR]] paper<ref>http://eprint.iacr.org/2011/232</ref> demonstrating that it is possible to retrieve a TLS private key of a server using [[OpenSSL]] that authenticates with Elliptic Curves DSA over a binary [[Field (mathematics)|field]] via a [[timing attack]].<ref>https://www.kb.cert.org/vuls/id/536044</ref> The vulnerability was fixed in OpenSSL 1.0.0e.<ref>http://www.openssl.org/news/changelog.html</ref>
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| In August 2013, it was revealed that bugs in some implementations of the [[Java (programming language)|Java]] class [http://docs.oracle.com/javase/7/docs/api/java/security/SecureRandom.html SecureRandom] sometimes generated collisions in the k value. As discussed above, this allowed solution of the private key, in turn allowing stealing [[bitcoin]]s from the containing wallet on [[Android (operating system)|Android]] app implementations, which use Java and rely on ECDSA to authenticate transactions.<ref>{{cite web |title=Android bug batters Bitcoin wallets
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| | url= http://www.theregister.co.uk/2013/08/12/android_bug_batters_bitcoin_wallets/ |publisher=The Register |date=12 August 2013}}</ref>
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| This issue can be prevented by deterministic generation of ''k'', as described by RFC 6979.
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| ==See also==
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| * [[Elliptic curve cryptography]]
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| * [[EdDSA]]
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| ==Notes==
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| <references />
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| ==References==
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| * Accredited Standards Committee [http://www.x9.org X9], ''American National Standard X9.62-2005, Public Key Cryptography for the Financial Services Industry, The Elliptic Curve Digital Signature Algorithm (ECDSA)'', November 16, 2005.
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| * Certicom Research, [http://www.secg.org/download/aid-780/sec1-v2.pdf ''Standards for efficient cryptography, SEC 1: Elliptic Curve Cryptography''], Version 2.0, May 21, 2009.
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| * López, J. and Dahab, R. [http://citeseer.ist.psu.edu/333066.html ''An Overview of Elliptic Curve Cryptography''], Technical Report IC-00-10, State University of Campinas, 2000.
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| * Daniel J. Bernstein, [http://cr.yp.to/papers/pippenger.pdf Pippenger's exponentiation algorithm], 2002.
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| * Daniel R. L. Brown, ''Generic Groups, Collision Resistance, and ECDSA'', Designs, Codes and Cryptography, '''35''', 119–152, 2005. [http://eprint.iacr.org/2002/026 ePrint version]
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| * Ian F. Blake, Gadiel Seroussi, and Nigel P. Smart, editors, ''Advances in Elliptic Curve Cryptography'', London Mathematical Society Lecture Note Series 317, Cambridge University Press, 2005.
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| * {{cite doi|10.1007/b97644}}
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| ==External links==
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| * [http://csrc.nist.gov/publications/fips/fips186-3/fips_186-3.pdf Digital Signature Standard; includes info on ECDSA]
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| {{Cryptography navbox | public-key}}
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| {{DEFAULTSORT:Elliptic Curve Dsa}}
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| [[Category:Digital signature schemes]]
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| [[Category:Elliptic curve cryptography]]
| |
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