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In physics, an '''operator''' is a [[Function (mathematics)|function]] acting on the space of physical states. As a result
'''A5/1''' is a [[stream cipher]] used to provide over-the-air communication [[privacy]] in the [[Global System for Mobile Communications|GSM]] [[Cell phone|cellular telephone]] standard.  It is one of seven algorithms which were specified for GSM use.<ref name=ChangeReq>{{cite web|url=http://www.3gpp.org/ftp/tsg_sa/TSG_SA/TSGS_37/Docs/SP-070671.zip|title=Prohibiting A5/2 in mobile stations and other clarifications regarding A5 algorithm support}}</ref> It was initially kept secret, but became public knowledge through leaks and [[reverse engineering]]. A number of serious weaknesses in the cipher have been identified.
of its application on a physical state, another physical state is obtained, very often along with
some extra relevant information.


The simplest example of the utility of operators is the study of [[symmetry]]. Because of this, they
==History and usage==
are a very useful tool in [[classical mechanics]]. In [[quantum mechanics]], on the other hand, they
A5/1 is used in [[Europe]] and the United States. [[A5/2]] was a deliberate weakening of the algorithm for certain export regions.<ref>{{cite web
are an intrinsic part of the formulation of the theory.
| archiveurl = http://web.archive.org/web/20040712061808/www.ausmobile.com/downloads/technical/Security+in+the+GSM+system+01052004.pdf
| first = Jeremy | last = Quirke | archivedate = 2004-07-12
| title = Security in the GSM system | publisher = AusMobile
| date = 2004-05-01 | url = http://www.ausmobile.com/downloads/technical/Security+in+the+GSM+system+01052004.pdf
}}</ref> A5/1 was developed in 1987, when GSM was not yet considered for use outside Europe, and [[A5/2]] was developed in 1989. Though both were initially kept secret, the general design was leaked in 1994 and the algorithms were entirely reverse engineered in 1999 by [[Marc Briceno]] from a GSM telephone. In 2000, around 130 million GSM customers relied on A5/1 to protect the confidentiality of their voice communications; by 2011, it was 4 billion.{{Citation needed|date=March 2011}}.


==Operators in classical mechanics==
Security researcher [[Ross J. Anderson|Ross Anderson]] reported in 1994 that "there was a terrific row between the [[NATO]] [[SIGINT|signal intelligence agencies]] in the mid-1980s over whether GSM encryption should be strong or not. The Germans said it should be, as they shared a long border with the [[Warsaw Pact]]; but the other countries didn't feel this way, and the algorithm as now fielded is a French design."<ref name="Ross94">{{cite newsgroup
| title = A5 (Was: HACKING DIGITAL PHONES)
| author = [[Ross J. Anderson|Ross Anderson]]
| date = 1994-06-17
| newsgroup = uk.telecom
|message-id= 2ts9a0$95r@lyra.csx.cam.ac.uk
| url = http://groups.google.com/groups?selm=2ts9a0%2495r%40lyra.csx.cam.ac.uk
}}</ref>


In classical mechanics, the dynamics of a particle (or system of particles) are completely determined by the [[Lagrangian]] ''L''(''q, q̇, t'') or equivalently the [[Hamiltonian mechanics|Hamiltonian]] ''H''(''q, p, t''), a function of the [[generalized coordinates]] ''q'', generalized velocities ''q̇'' = d''q''/d''t'' and its [[conjugate momenta]]:
==Description==
[[Image:A5-1 GSM cipher.svg|thumbnail|350px|right|The A5/1 stream cipher uses three LFSRs. A register is clocked if its clocking bit (orange) agrees with one or both of the clocking bits of the other two registers.]]
A GSM transmission is organised as sequences of ''bursts''. In a typical channel and in one direction, one burst is sent every 4.615 milliseconds and contains 114 bits available for information. A5/1 is used to produce for each burst a 114 bit sequence of [[keystream]] which is [[XOR]]ed with the 114 bits prior to modulation. A5/1 is initialised using a 64-bit [[key (cryptography)|key]] together with a publicly known 22-bit frame number. Older fielded GSM implementations using Comp128v1 for key generation, had 10 of the key bits fixed at zero, resulting in an effective [[key length]] of 54 bits. This weakness was rectified with the introduction of Comp128v2 which yields proper 64 bits keys. When operating in GPRS / EDGE mode, higher bandwidth radio modulation allows for larger 348 bits frames, and [[A5/3]] is then used in a stream cipher mode to maintain confidentiality.


:<math>p = \frac{\partial L}{\partial \dot{q}}</math>
A5/1 is based around a combination of three [[linear feedback shift register]]s (LFSRs) with irregular clocking. The three shift registers are specified as follows:
 
{| class="wikitable"
If either ''L'' or ''H'' are independent of a generalized coordinate ''q'', meaning the ''L'' and ''H'' so not change when ''q'' is changed, which in turn means the dynamics of the particle are still the same even when ''q'' changes, the corresponding momenta conjugate to those coordinates will be conserved (this is part of [[Noether's theorem]], and the invariance of motion with respect to the coordinate ''q'' is a [[symmetry (physics)|symmetry]]). Operators in classical mechanics are related to these symmetries.
!LFSR<br>number
 
!Length in <br>bits
More technically, when ''H'' is invariant under the action of a certain [[group (mathematics)|group]] of transformations ''G'':  
!Feedback<br> polynomial
 
!Clocking <br>bit
:<math>S\in G, H(S(q,p))=H(q,p)</math>.
!Tapped <br>bits
 
the elements of ''G'' are physical operators, which map physical states among themselves.
 
===Table of classical mechanics operators===
 
:{| class="wikitable"
|-
! Transformation
! Operator
! Position
! Momentum
|-
| [[Translational symmetry]]
| <math>X(\bold{a})</math>
| <math>\bold{r}\rightarrow \bold{r} + \bold{a}</math>
| <math>\bold{p}\rightarrow \bold{p}</math>
|-
|-
| [[Time evolution|Time translations]]
|1 || 19 || <math>x^{19} + x^{18} + x^{17} + x^{14} + 1</math> || 8 || 13, 16, 17, 18
| <math>U(t_0)</math>
| <math>\bold{r}(t)\rightarrow \bold{r}(t+t_0)</math>
| <math>\bold{p}(t)\rightarrow \bold{p}(t+t_0)</math>
|-
|-
| [[Rotational invariance]]
|2 || 22 || <math>x^{22} + x^{21} + 1</math> || 10 || 20, 21
| <math>R(\bold{\hat{n}},\theta)</math>
| <math>\bold{r}\rightarrow R(\bold{\hat{n}},\theta)\bold{r}</math>
| <math>\bold{p}\rightarrow R(\bold{\hat{n}},\theta)\bold{p}</math>
|-
| [[Galilean transformation]]s
| <math>G(\bold{v})</math>
| <math>\bold{r}\rightarrow \bold{r} + \bold{v}t</math>
| <math>\bold{p}\rightarrow \bold{p} + m\bold{v}</math>
|-
| [[Parity (physics)|Parity]]
| <math>P</math>
| <math>\bold{r}\rightarrow -\bold{r}</math>
| <math>\bold{p}\rightarrow -\bold{p}</math>
|-
| [[T-symmetry]]
| <math>T</math>
| <math>\bold{r}\rightarrow \bold{r}(-t)</math>
| <math>\bold{p}\rightarrow -\bold{p}(-t)</math>
|-
|-
|3 || 23 || <math>x^{23} + x^{22} + x^{21} + x^{8} + 1</math> || 10 || 7, 20, 21, 22
|}
|}
The bits are indexed with the [[least significant bit]] (LSB) as 0.


where ''R''('''n̂''', θ) is the [[rotation matrix]] about an axis defined by the [[unit vector]] '''n̂''' and angle θ.
The registers are clocked in a stop/go fashion using a majority rule. Each register has an associated clocking bit. At each cycle, the clocking bit of all three registers is examined and the majority bit is determined. A register is clocked if the clocking bit agrees with the majority bit. Hence at each step at least two or three registers are clocked, and each register steps with probability 3/4.


==Concept of generator==
Initially, the registers are set to zero. Then for 64 cycles, the 64-bit secret key is mixed in according to the following scheme: in cycle <math>0\leq{i}<64</math>, the ''i''th key bit is added to the least significant bit of each register using XOR —
If the transformation is infinitesimal, the operator action should be of the form
:<math>R[0] = R[0] \oplus K[i].</math>
Each register is then clocked.


: <math> I + \epsilon A </math>
Similarly, the 22-bits of the frame number are added in 22 cycles. Then the entire system is clocked using the normal majority clocking mechanism for 100 cycles, with the output discarded. After this is completed, the cipher is ready to produce two 114 bit sequences of output keystream, first 114 for downlink, last 114 for uplink.


where <math>I</math> is the identity operator, <math>\epsilon</math> is a small parameter, and <math>A</math> will depend on the transformation at hand, and is called a generator of the group. Again, as a simple example, we will derive the generator of the space translations on 1D functions.
==Security==
[[File:CipheringNotProvided.jpg|right|thumb|200 px|The message on the screen of a mobile phone with the warning about lack of ciphering]]A number of attacks on A5/1 have been published, and the American [[National Security Agency]] is able to routinely decrypt A5/1 messages according to released internal documents.<ref>[http://yro.slashdot.org/story/13/12/14/0148251/nsa-able-to-crack-a51-cellphone-crypto NSA Able To Crack A5/1 Cellphone Crypto - Slashdot]</ref>


As it was stated, <math>T_a f(x)=f(x-a)</math>. If <math>a=\epsilon</math> is infinitesimal, then we may  write
Some attacks require an expensive preprocessing stage after which the cipher can be broken in minutes or seconds. Until recently, the weaknesses have been passive attacks using the [[known plaintext]] assumption. In 2003, more serious weaknesses were identified which can be exploited in the [[ciphertext only attack|ciphertext-only scenario]], or by an active attacker. In 2006 Elad Barkan, [[Eli Biham]] and Nathan Keller demonstrated attacks against A5/1, [[A5/3]], or even GPRS that allow attackers to tap GSM mobile phone conversations and decrypt them either in real-time, or at any later time.


: <math>T_\epsilon f(x)=f(x-\epsilon)\approx f(x) - \epsilon f'(x).</math>
According to professor Jan Arild Audestad, at the standardization process which started in 1982, A5/1 was originally proposed to have a key length of 128 bits. At that time, 128 bits was projected to be secure for at least 15 years. It is now estimated that 128 bits would in fact also still be secure as of 2014. Audestad, Peter van der Arend, and [[Thomas Haug]] says that the British insisted on weaker encryption, with Haug saying he was told by the British delegate that this was to allow the British secret service to eavesdrop more easily. The British proposed a key length of 48 bits, while the West Germans wanted stronger encryption to protect against East German spying, so the compromise became a key length of 56 bits.<ref>http://www.aftenposten.no/nyheter/uriks/Sources-We-were-pressured-to-weaken-the-mobile-security-in-the-80s-7413285.html#.UtBeNpD_sQs</ref>


This formula may be rewritten as
===Known-plaintext attacks===
The first attack on the A5/1 was proposed by [[Ross J. Anderson|Ross Anderson]] in 1994. Anderson’s basic idea was to guess the complete content of the registers R1 and R2 and about half of the register R3. In this way the clocking of all three registers is determined and the second half of R3 can be computed.<ref name="Ross94"/>


: <math>T_\epsilon f(x) = (I-\epsilon D) f(x)</math>
In 1997, Golic presented an attack based on solving sets of linear equations which has a time complexity of 2<sup>40.16</sup> (the units are in terms of number of solutions of a system of linear equations which are required).


where <math>D</math> is the generator of the translation group, which in this case happens to be the ''derivative'' operator. Thus, it is said that the generator of translations is the derivative.
In 2000, [[Alex Biryukov]], [[Adi Shamir]] and [[David A. Wagner|David Wagner]] showed that A5/1 can be [[cryptanalysis|cryptanalysed]] in real time using a time-memory tradeoff attack,<ref>{{cite journal
| authorlink = Alex Biryukov | first = Alex | last =  Biryukov |author2=  [[Adi Shamir]]|author3= [[David A. Wagner|David Wagner]]
| title =  Real Time Cryptanalysis of A5/1 on a PC
| journal =  [[Fast Software Encryption]]—FSE 2000 | pages = 1–18 | url = http://cryptome.info/0001/a51-bsw/a51-bsw.htm
}}</ref> based on earlier work by Jovan Golic.<ref>{{cite journal
| first = Jovan Dj. | last =  Golic | title =  Cryptanalysis of Alleged A5 Stream Cipher
| work = [[EUROCRYPT]] 1997 | year = 1997 | pages = 239–55 | url = http://jya.com/a5-hack.htm
}}</ref> One tradeoff allows an attacker to reconstruct the key in one second from two minutes of known plaintext or in several minutes from two seconds of known plain text, but he must first complete an expensive preprocessing stage which requires 2<sup>48</sup> steps to compute around 300 GB of data. Several tradeoffs between preprocessing, data requirements, attack time and memory complexity are possible.


==The exponential map==
The same year, [[Eli Biham]] and [[Orr Dunkelman]] also published an attack on A5/1 with a total work complexity of 2<sup>39.91</sup> A5/1 clockings given 2<sup>20.8</sup> bits of [[known plaintext]]. The attack requires 32 GB of data storage after a [[precomputation]] stage of 2<sup>38</sup>.<ref>{{cite journal
The whole group may be recovered, under normal circumstances, from the generators, via the [[exponential map]]. In the case of the translations the idea works like this.
| first = Eli | last = Biham |author2=  Orr Dunkelman
| title = Cryptanalysis of the A5/1 GSM Stream Cipher | year = 2000
| journal = [[Indocrypt]] 2000 | pages =43–51
}}</ref>


The translation for a finite value of <math>a</math> may be obtained by repeated application of the infinitesimal translation:
Ekdahl and Johannson published an attack on the initialisation procedure which breaks A5/1 in a few minutes using two to five minutes of conversation plaintext.<ref>{{cite journal
| first = Patrik | last =  Ekdahl |author2= Thomas Johansson
| title  = Another attack on A5/1
| journal = IEEE Transactions on Information Theory | volume = 49 | issue = 1
| pages = 284–89 | year = 2003 | url = http://www.it.lth.se/patrik/papers/a5full.pdf (
| doi = 10.1109/TIT.2002.806129
}}</ref> This attack does not require a preprocessing stage. In 2004, Maximov et al. improved this result to an attack requiring "less than one minute of computations, and a few seconds of known conversation". The attack was further improved by [[Elad Barkan]] and [[Eli Biham]] in 2005.<ref>{{cite journal
| first = Elad | last =  Barkan |author2= Eli Biham
| title =  Conditional Estimators: An Effective Attack on A5/1
| journal =  Selected Areas in Cryptography 2005 | year = 2005 | pages = 1–19
}}</ref>


: <math>T_a f(x) = \lim_{N\to\infty} T_{a/N} \cdots T_{a/N} f(x)</math>
===Attacks on A5/1 as used in GSM===


with the <math>\cdots</math> standing for the application <math>N</math> times. If <math>N</math> is large, each of the factors may be considered to be infinitesimal:
In 2003, Barkan ''et al.'' published several attacks on GSM encryption.<ref>{{cite journal
| first = Elad | last = Barkan |author2= [[Eli Biham]]|author3=  Nathan Keller
| title =  Instant Ciphertext-Only Cryptanalysis of GSM Encrypted Communication | year = 2003
| journal =  [[Crypto]] 2003 | pages = 600–16  | url = http://cryptome.org/gsm-crack-bbk.pdf
}}</ref> The first is an active attack. GSM phones can be convinced to use the much weaker [[A5/2]] cipher briefly. A5/2 can be broken easily, and the phone uses the same key as for the stronger A5/1 algorithm. A second attack on A5/1 is outlined, a [[ciphertext-only]] time-memory tradeoff attack which requires a large amount of precomputation.


: <math>T_a f(x) = \lim_{N\to\infty} (I -(a/N) D)^N f(x).</math>
In 2006, [[Elad Barkan]], [[Eli Biham]], [[Nathan Keller]] published the full version of their 2003 paper, with attacks against A5/X Ciphers. The authors claim:<ref>{{cite web
| url = http://www.cs.technion.ac.il/users/wwwb/cgi-bin/tr-get.cgi/2006/CS/CS-2006-07.pdf
| title = Instant Ciphertext-Only Cryptanalysis of GSM Encrypted Communication by Barkan and Biham of Technion (Full Version)
| first = Elad | last = Barkan |author2= Eli Biham|author3= Nathan Keller
}}</ref> {{quotation|We present a very practical ciphertext-only cryptanalysis of GSM encrypted communication, and various active attacks on the GSM protocols. These attacks can even break into GSM networks that use "unbreakable" ciphers. We first describe a ciphertext-only attack on A5/2 that requires a few dozen milliseconds of encrypted off-the-air cellular conversation and finds the correct key in less than a second on a personal computer. We extend this attack to a (more complex) ciphertext-only attack on A5/1. We then describe new (active) attacks on the protocols of networks that use A5/1, A5/3, or even GPRS. These attacks exploit flaws in the GSM protocols, and they work whenever the mobile phone supports a weak cipher such as A5/2. We emphasize that these attacks are on the protocols, and are thus applicable whenever the cellular phone supports a weak cipher, for example, they are also applicable for attacking A5/3 networks using the cryptanalysis of A5/1. Unlike previous attacks on GSM that require unrealistic information, like long known plaintext periods, our attacks are very practical and do not require any knowledge of the content of the conversation. Furthermore, we describe how to fortify the attacks to withstand reception errors. As a result, our attacks allow attackers to tap conversations and decrypt them either in real-time, or at any later time.}}


But this limit may be rewritten as an exponential:
In 2007 [[University of Bochum|Universities of Bochum]] and Kiel started a research project to create a massively parallel [[FPGA]]-based cryptographic accelerator [http://www.sciengines.com/copacobana/ COPACOBANA]. COPACOBANA was the first commercially available solution<ref>{{cite journal
| first = Tim | last =  Gueneysu |author2= Timo Kasper|author3= Martin Novotný|author4= Christof Paar|author5= Andy Rupp
| title = Cryptanalysis with COPACOBANA
| url = http://www.sciengines.com/copacobana/paper/TC_COPACOBANA.pdf
| journal =  [[Transactions on Computers]] Nov. 2008 | year = 2008 | pages = 1498–1513
| volume = 57
}}</ref> using fast time-memory trade-off techniques that could be used to attack the popular A5/1 and A5/2 algorithms, used in GSM voice encryption, as well as the [[Data Encryption Standard]] (DES). It also enables [[brute force attack]]s against GSM eliminating the need of large precomputated lookup tables.


: <math>T_a f(x)= \exp(-aD) f(x).</math>
In 2008, the group [[The Hackers Choice]] launched a project to develop a practical attack on A5/1. The attack requires the construction of a large look-up table of approximately 3 terabytes.  Together with the scanning capabilities developed as part of the sister project, the group expected to be able to record any GSM call or SMS encrypted with A5/1, and within about 3–5 minutes derive the encryption key and hence listen to the call and read the SMS in clear.  But the tables weren't released.<ref name="nohl26c3"/>


To be convinced of the validity of this formal expression, we may expand the exponential in a power series:
A similar effort, the [http://reflextor.com/trac/a51/wiki A5/1 Cracking Project], was announced at the [[Black Hat Briefings|2009 Black Hat security conference]] by cryptographers [[:de:Karsten Nohl|Karsten Nohl]] and Sascha Krißler.  It created the look-up tables using [[Nvidia]] [[GPGPU]]s via a [[peer-to-peer]] [[distributed computing]] architecture.  Starting in the middle of September 2009, the project ran the equivalent of 12 Nvidia GeForce GTX 260. According to the authors, the approach can be used on any cipher with key size up to 64-bits.<ref name=nohl/>


: <math>T_a f(x) = \left( I - aD + {a^2D^2\over 2!} - {a^3D^3\over 3!} + \cdots \right) f(x).</math>
In December 2009, the A5/1 Cracking Project attack tables for A5/1 were announced by Chris Paget and Karsten Nohl. The tables use a combination of compression techniques, including [[rainbow table]]s and distinguished point chains. These tables constituted only parts of the 2TB completed table, and had been computed during three months using 40 distributed [[CUDA]] nodes, and then published over [[BitTorrent (protocol)|BitTorrent]].<ref name="nohl26c3">{{Cite conference
| conference = 26th Chaos Communication Congress  (26C3):
| last = Nohl
| first = Karsten
|author2= Chris Paget
| title = GSM: SRSLY?
| accessdate = 2009-12-30
| date = 2009-12-27
| url = http://events.ccc.de/congress/2009/Fahrplan/events/3654.en.html
| archiveurl= http://web.archive.org/web/20100106084817/http://events.ccc.de/congress/2009/Fahrplan/events/3654.en.html| archivedate= 6 January 2010 <!--DASHBot-->| deadurl= no}}</ref><ref name=nohl>https://har2009.org/program/attachments/119_GSM.A51.Cracking.Nohl.pdf Subverting the security base of GSM. Karsten Nohl and Sascha Krißler</ref><ref>{{cite news|url=http://www.nytimes.com/2009/12/29/technology/29hack.html|title=Cellphone Encryption Code Is Divulged|first=Kevin|last=O'Brien|date=2009-12-28|accessdate=2009-12-29|work=New York Times| archiveurl= http://web.archive.org/web/20100101130022/http://www.nytimes.com/2009/12/29/technology/29hack.html| archivedate= 1 January 2010 <!--DASHBot-->| deadurl= no}}</ref><ref>{{cite news |url=http://www.pcworld.com/article/185542/hackers_show_its_easy_to_snoop_on_a_gsm_call.html |title=Hackers Show It's Easy to Snoop on a GSM Call |first=Robert |last=McMillan |work=IDG News Service }}</ref> More recently the project has announced a switch to faster ATI [[Evergreen (GPU family)|Evergreen]] code, together with a change in the format of the tables and [[Frank A. Stevenson]] announced [http://web.archive.org/web/20120306125406/http://lists.lists.reflextor.com/pipermail/a51/2010-May/000605.html breaks of A5/1] using the ATI generated tables.


The right-hand side may be rewritten as
Documents leaked by Edward Snowden in 2013 states that NSA "can process encrypted A5/1".<ref>http://www.washingtonpost.com/business/technology/by-cracking-cellphone-code-nsa-has-capacity-for-decoding-private-conversations/2013/12/13/e119b598-612f-11e3-bf45-61f69f54fc5f_story.html</ref>


: <math>f(x) - a f'(x) + {a^2\over 2!} f''(x) - {a^3\over 3!} f'''(x) + \cdots</math>
==See also==
 
* [[A5/2]]
which is just the Taylor expansion of <math>f(x-a)</math>, which was our original value for <math>T_a f(x)</math>.
* [[KASUMI (block cipher)|KASUMI]], also known as A5/3
 
* [[Cellular Message Encryption Algorithm]]
The mathematical properties of physical operators are a topic of great importance in itself. For further information, see [[C*-algebra]] and [[Gelfand-Naimark theorem]].
 
==Operators in quantum mechanics==
 
The [[mathematical formulation of quantum mechanics]] (QM) is built upon the concept of an operator.
 
The wavefunction represents the [[probability amplitude]] of finding the system in that state. The terms "wavefunction" and "state" in QM context are usually used interchangeably.
 
Physical [[pure state]]s in quantum mechanics are represented as [[unit-norm vector]]s (probabilities are normalized to one) in a special [[complex number|complex]] [[vector space]]: a [[Hilbert space]]. [[Time evolution]] in this vector space is given by the application of the [[evolution operator]].
 
Any [[observable]], i.e., any quantity which can be measured in a physical experiment, should be associated with a [[self-adjoint]] [[linear operator]]. The operators must yield real [[eigenvalue]]s, since they are values which may come up as the result of the experiment. Mathematically this means the operators must be [[Hermitian matrix|Hermitian]].<ref>Molecular Quantum Mechanics Parts I and II: An Introduction to QUANTUM CHEMISRTY (Volume 1), P.W. Atkins, Oxford University Press, 1977, ISBN 0-19-855129-0</ref> The probability of each eigenvalue is related to the projection of the physical state on the subspace related to that eigenvalue. See below for mathematical details.
 
In the [[wave mechanics]] formulation of QM, the wavefunction varies with space and time, or equivalently momentum and time (see [[position and momentum space]] for details), so observables are [[differential operator]]s.
 
In the [[matrix mechanics]] formulation, the [[Norm (mathematics)|norm]] of the physical state should stay fixed, so the evolution operator should be [[unitary transformation|unitary]], and the operators can be represented as matrices. Any other symmetry, mapping a physical state into another, should keep this restriction.
 
===Wavefunction ===
 
{{Main|wavefunction}}
 
The wavefunction must be [[square-integrable]] on the Hilbert space (see [[Lp spaces]]) meaning:
 
:<math>\int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty |\psi(\bold{r})|^2 {\rm d}^3\bold{r} = \int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty \psi(\bold{r})^*\psi(\bold{r}){\rm d}^3\bold{r} < \infty </math>
 
and normalizable, so that:
 
:<math>\int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty |\psi(\bold{r})|^2 {\rm d}^3\bold{r} = 1 </math>
 
Two cases of eigenstates (and eigenvalues) are:
*for '''discrete''' eigenstates <math> | \psi_i \rangle </math> forming a discrete basis, so the state is a [[sum]]
::<math>|\psi\rangle = \sum_i c_i|\phi_i\rangle</math>
:where ''c<sub>i</sub>'' are complex numbers such that |''c<sub>i</sub>''|<sup>2</sup> = ''c<sub>i</sub>''<sup>*</sup>''c<sub>i</sub>'' = probability of measuring the state <math>|\phi_i\rangle</math>, and has the corresponding set of eigenvalues ''a<sub>i</sub>'' is also discrete - either [[finite]] or [[countably infinite]],
*for a '''continuum''' of eigenstates <math> | \psi \rangle </math> forming a continuous basis, so the state is an [[integral]]
::<math>|\psi\rangle = \int c(\phi){\rm d}\phi|\phi_i\rangle </math>
:where ''c''(φ) is a complex function such that |''c''(φ)|<sup>2</sup> = ''c''(φ)<sup>*</sup>''c''(φ) = probability of measuring the state <math>|\phi\rangle</math>, there is an [[uncountably infinite]] set of eigenvalues ''a''.
 
===Linear operators in wave mechanics===
 
{{Main|Wave function|Bra-ket notation}}
 
Let ''ψ'' be the wavefunction for a quantum system, and <math>\hat{A}</math> be any [[linear operator]] for some observable ''A'' (such as position, momentum, energy, angular momentum etc.), then
 
:<math>\hat{A} \psi = a \psi ,</math>
 
where:
 
* ''a'' is the [[Eigenvalues and eigenvectors|eigenvalue]] of the operator, corresponding to the measured value of the observable, i.e. observable ''A'' has a measured value ''a''
*''ψ'' is the [[eigenfunction]] of <math>\hat{A}</math> if this relation holds.
 
If ''ψ''  is an eigenfunction of an operator, it means the eigenvalue can be found and so the observable can be measured, conversely if is not an eigenfunction then the eigenvalue can't be found and the observable can't be measured for that case.
 
In bra-ket notation the above can be written;
 
:<math>\begin{align} & \hat{A} \psi = \hat{A} \psi ( \mathbf{r} ) = \hat{A} \langle \mathbf{r} | \psi \rangle = \langle \mathbf{r} | \hat {A} | \psi \rangle \\
& a \psi = a \psi ( \mathbf{r} ) = a \langle \mathbf{r} | \psi \rangle = \langle \mathbf{r} | a | \psi \rangle \\
\end{align} </math>
 
in which case <math> | \psi \rangle </math> is an [[eigenvector]], or [[eigenket]].
 
Due to linearity, vectors can be defined in any number of dimensions, as each component of the vector acts on the function separately. One mathematical example is the [[del operator]], which is itself a vector (useful in momentum-related quantum operators, in the table below).
 
An operator in ''n''-dimensional space can be written:
 
:<math> \mathbf{\hat{A}} = \sum_{j=1}^n \mathbf{e}_\mathrm{j} \hat{A}_j </math>
 
where '''e'''<sub>''j''</sub> are basis vectors corresponding to each component operator ''A<sub>j</sub>''. Each component will yield a corresponding eigenvalue. Acting this on the wave function ''ψ'':
 
:<math> \mathbf{\hat{A}} \psi = \left ( \sum_{j=1}^n \mathbf{e}_\mathrm{j} \hat{A}_j \right ) \psi = \sum_{j=1}^n \left ( \mathbf{e}_\mathrm{j} \hat{A}_j \psi \right ) = \sum_{j=1}^n \left ( \mathbf{e}_\mathrm{j} a_j \psi \right ) </math>
 
in which
 
:<math> \hat{A}_j \psi = a_j \psi .</math>
 
In bra-ket notation:
 
:<math>\begin{align} & \mathbf{\hat{A}} \psi = \mathbf{\hat{A}} \psi ( \mathbf{r} ) = \mathbf{\hat{A}} \langle \mathbf{r} | \psi \rangle = \langle \mathbf{r} | \mathbf{\hat{A}} | \psi \rangle \\
 
& \left ( \sum_{j=1}^n \mathbf{e}_\mathrm{j} \hat{A}_j \right ) \psi = \left ( \sum_{j=1}^n \mathbf{e}_\mathrm{j} \hat{A}_j \right ) \psi ( \mathbf{r} ) = \left ( \sum_{j=1}^n \mathbf{e}_\mathrm{j} \hat{A}_j \right ) \langle \mathbf{r} | \psi \rangle = \left \langle \mathbf{r} \Bigg | \sum_{j=1}^n \mathbf{e}_\mathrm{j} \hat{A}_j \Bigg | \psi \right \rangle \\
 
\end{align} \,\!</math>
 
===Commutation of operators on ''Ψ''===
 
{{main|Commutator}}
 
If two observables ''A'' and ''B'' have linear operators <math> \hat{A} </math> and <math> \hat{B} </math>, the commutator is defined by,
 
:<math> \left [ \hat{A}, \hat{B} \right ] = \hat{A} \hat{B} - \hat{B} \hat{A} </math>
 
The commutator is itself a (composite) operator. Acting the commutator on ''ψ'' gives:
 
:<math> \left [ \hat{A}, \hat{B} \right ] \psi = \hat{A} \hat{B} \psi - \hat{B} \hat{A} \psi . </math>
 
If ''ψ'' is an eigenfunction with eigenvalues ''a'' and ''b'' for observables ''A'' and ''B'' respectively, and if the operators commute:
 
:<math> \left [ \hat{A}, \hat{B} \right ] \psi = 0, </math>
 
then the observables ''A'' and ''B'' can be measured at the same time with measurable eigenvalues ''a'' and ''b'' respectively. To illustrate this:
 
:<math> \begin{align}\left [ \hat{A}, \hat{B} \right ] \psi & = \hat{A} \hat{B} \psi - \hat{B} \hat{A} \psi \\
& = a(b \psi) - b(a \psi) \\
& = 0 .\\
\end{align} </math>
 
If the operators do not commute:
 
:<math> \left [ \hat{A}, \hat{B} \right ] \psi \neq 0, </math>
 
they can't be measured simultaneously to arbitrary precision, and there is an uncertainty relation between the observables, even if ''ψ''  is an eigenfunction. Notable pairs are position and momentum, and energy and time - [[Uncertainty principle|Hiesenberg's uncertainty relations]], and the angular momenta (spin, orbital and total) about any two orthogonal axes (such as ''L<sub>x</sub>'' and ''L<sub>y</sub>'', or ''s<sub>y</sub>'' and ''s<sub>z</sub>'' etc.).
 
===Expectation values of operators on ''Ψ''===
 
The [[expectation value]] (equivalently the average or mean value) is the average measurement of an observable, for particle in region ''R''. The expectation value <math>\langle \hat{A} \rangle </math> of the operator <math> \hat{A} </math> is calculated from<ref>Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10) 0 07 145546 9</ref>:
 
:<math>\langle \hat{A} \rangle = \int_R \psi^{*}\left( \mathbf{r} \right ) \hat{A} \psi \left( \mathbf{r} \right ) \mathrm{d}^3\mathbf{r} = \langle \psi | \hat{A} | \psi \rangle .</math>
 
This can be generalized to any function ''F'' of an operator:
 
:<math> \langle F ( \hat{A} ) \rangle = \int_R \psi(\mathbf{r})^{*} \left [ F ( \hat{A} ) \psi(\mathbf{r}) \right ] \mathrm{d}^3 \mathbf{r} = \langle \psi | F ( \hat{A} ) | \psi \rangle , </math>
 
An example of ''F'' is the 2-fold action of ''A'' on ''ψ'', i.e. squaring an operator or doing it twice:
 
:<math>\begin{align}
& F(\hat{A}) = \hat{A}^2 \\
& \Rightarrow \langle \hat{A}^2 \rangle = \int_R \psi^{*} \left( \mathbf{r} \right ) \hat{A}^2 \psi \left( \mathbf{r} \right ) \mathrm{d}^3\mathbf{r} = \langle \psi \vert \hat{A}^2 \vert \psi \rangle \\
\end{align}\,\!</math>
 
===Hermitian operators===
 
{{Main|Self-adjoint operator}}
 
The definition of a [[Hermitian operator]] is <ref>Molecular Quantum Mechanics Parts I and II: An Introduction to QUANTUM CHEMISRTY (Volume 1), P.W. Atkins, Oxford University Press, 1977, ISBN 0-19-855129-0</ref>:
 
:<math>\hat{A} = \hat{A}^\dagger</math>
 
Following from this, in bra-ket notation:
 
:<math>\langle \phi_i | \hat{A} | \phi_j \rangle = \langle \phi_j | \hat{A} | \phi_i \rangle^*.</math>
 
Important properties of Hermitian operators include:
 
*real eigenvalues,
*eigenvectors with different eigenvalues are [[orthogonal]],
*eigenvectors can be chosen to be a complete [[orthonormal basis]],
 
===Operators in Matrix mechanics ===
 
An operator can be written in matrix form to map one basis vector to another. Since the operators and basis vectors are linear, the matrix is a [[linear transformation]] (aka transition matrix) between bases. Each basis element <math>\phi_j </math> can be connected to another<ref>Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10) 0 07 145546 9</ref>, by the expression:
 
:<math>A_{ij} = \langle \phi_i | \hat{A} | \phi_j \rangle,</math>
 
which is a matrix element:
 
:<math>\hat{A} = \begin{pmatrix}
A_{11} & A_{12} & \cdots & A_{1n} \\
A_{21} & A_{22} & \cdots & A_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
A_{n1} & A_{n2} & \cdots & A_{nn} \\
\end{pmatrix}
</math>
 
A further property of a Hermitian operator is that eigenfunctions corresponding to different eigenvalues are orthogonal.<ref>Molecular Quantum Mechanics Parts I and II: An Introduction to QUANTUM CHEMISRTY (Volume 1), P.W. Atkins, Oxford University Press, 1977, ISBN 0-19-855129-0</ref> In matrix form, operators allow real eigenvalues to be found, corresponding to measurements. Orthogonality allows a suitable basis set of vectors to represent the state of the quantum system. The eigenvalues of the operator are also evaluated in the same way as for the square matrix, by solving the [[characteristic polynomial]]:
 
:<math> \det\left ( \hat{A} - a \hat{I} \right ) = 0 ,</math>
 
where ''I'' is the ''n'' × ''n'' [[identity matrix]], as an operator it corresponds to the identity operator. For a discrete basis:
 
:<math> \hat{I} = \sum_i |\phi_i\rangle\langle\phi_i|</math>
 
while for a continuous basis:
 
:<math> \hat{I} = \int |\phi\rangle\langle\phi|d\phi</math>
 
=== Inverse of an operator ===
 
A non-singular operator <math>\hat{A}</math> has an inverse <math> \hat{A}^{-1} </math> defined by:
 
:<math> \hat{A}\hat{A}^{-1} = \hat{A}^{-1}\hat{A} = \hat{I} </math>
 
If an operator has no inverse, it is a singular operator. In a finite-dimensional space, the determinant of a non-singular operator is non-zero:
 
:<math> \det(\hat{A}) \neq 0</math>
 
and hence it is zero for a singular operator.
 
===Table of QM operators===
 
The operators used in quantum mechanics are collected in the table below (see for example,<ref>Molecular Quantum Mechanics Parts I and II: An Introduction to QUANTUM CHEMISRTY (Volume 1), P.W. Atkins, Oxford University Press, 1977, ISBN 0-19-855129-0</ref><ref>Quanta: A handbook of concepts, P.W. Atkins, Oxford University Press, 1974, ISBN 0-19-855493-1</ref>). The bold-face vectors with circumflexes are not [[unit vector]]s, they are 3-vector operators; all three spatial components taken together.
 
:{| class="wikitable"
|-valign="top"
 
! scope="col" width="200" | Operator (common name/s)
! scope="col" width="200" | Cartesian component
! scope="col" width="200" | General definition
! scope="col" width="100" | SI unit
! scope="col" width="100" | Dimension
|-valign="top"
! [[Position operator|Position]]
|<math>\begin{align} \hat{x} = x \\
\hat{y} = y \\
\hat{z} = z
\end{align}</math>
|<math> \mathbf{\hat{r}} = \mathbf{r} \,\!</math>
| m
| [L]
|-valign="top"
!rowspan="2"| [[Momentum operator|Momentum]]
|General
 
<math> \begin{align}
\hat{p}_x & = -i \hbar \frac{\partial }{\partial x} \\
\hat{p}_y & = -i \hbar \frac{\partial }{\partial y} \\
\hat{p}_z & = -i \hbar \frac{\partial }{\partial z}
\end{align}</math>
|General
 
<math> \mathbf{\hat{p}} = -i \hbar \nabla \,\!</math>
| J s m<sup>−1</sup> = N s
| [M] [L] [T]<sup>−1</sup>
|-valign="top"
|Electromagnetic field
 
<math> \begin{align}
\hat{p}_x = -i \hbar \frac{\partial }{\partial x} - qA_x \\
\hat{p}_y = -i \hbar \frac{\partial }{\partial y} - qA_y \\
\hat{p}_z = -i \hbar \frac{\partial }{\partial z} - qA_z
\end{align}</math>
|Electromagnetic field (uses [[kinetic momentum]], '''A''' = vector potential)


<math> \begin{align}
==Notes==
\mathbf{\hat{p}} & = \bold{\hat{P}} - q\bold{A} \\
{{Reflist|2}}
& = -i \hbar \nabla - q\bold{A} \\
\end{align}\,\!</math>
| J s m<sup>−1</sup> = N s
| [M] [L] [T]<sup>−1</sup>
|-valign="top"
!rowspan="3"| [[Kinetic energy]]
| Translation
<math> \begin{align} \hat{T}_x & = -\frac{\hbar^2}{2m}\frac{\partial^2 }{\partial x^2} \\
\hat{T}_y & = -\frac{\hbar^2}{2m}\frac{\partial^2 }{\partial y^2} \\
\hat{T}_z & = -\frac{\hbar^2}{2m}\frac{\partial^2 }{\partial z^2} \\
\end{align} </math>
|
<math> \begin{align} \hat{T} & = \frac{\mathbf{\hat{p}}\cdot\mathbf{\hat{p}}}{2m} \\
& = \frac{(-i \hbar \nabla)\cdot(-i \hbar \nabla)}{2m} \\
& = \frac{-\hbar^2 }{2m}\nabla^2
\end{align}\,\!</math>
| J
| [M] [L]<sup>2</sup> [T]<sup>−2</sup>
|-valign="top"
|Electromagnetic field
 
<math> \begin{align} \hat{T}_x & = \frac{1}{2m}\left(-i \hbar \frac{\partial }{\partial x } - q A_x \right)^2 \\
\hat{T}_y & = \frac{1}{2m}\left(-i \hbar \frac{\partial }{\partial y} - q A_y \right)^2 \\
\hat{T}_z & = \frac{1}{2m}\left(-i \hbar \frac{\partial }{\partial z} - q A_z \right)^2
\end{align}\,\!</math>
|Electromagnetic field ('''A''' = [[vector potential]])
 
<math> \begin{align} \hat{T} & = \frac{\mathbf{\hat{p}}\cdot\mathbf{\hat{p}}}{2m} \\
& = \frac{1}{2m}(-i \hbar \nabla - q\bold{A})\cdot(-i \hbar \nabla - q\bold{A}) \\
& = \frac{1}{2m}(-i \hbar \nabla - q\bold{A})^2
\end{align}\,\!</math>
| J
| [M] [L]<sup>2</sup> [T]<sup>−2</sup>
|-valign="top"
|Rotation (''I'' = [[moment of inertia]])
 
<math> \begin{align}
\hat{T}_{xx} & = \frac{\hat{J}_x^2}{2I_{xx}} \\
\hat{T}_{yy} & = \frac{\hat{J}_y^2}{2I_{yy}} \\
\hat{T}_{zz} & = \frac{\hat{J}_y^2}{2I_{zz}} \\
\end{align}\,\!</math>
|Rotation
 
<math> \hat{T} = \frac{\bold{\hat{J}}\cdot\bold{\hat{J}}}{2I} \,\!</math>
| J
| [M] [L]<sup>2</sup> [T]<sup>−2</sup>
|-valign="top"
! Potential energy
| N/A
|<math> \hat{V} = V\left ( \mathbf{r}, t \right ) = V \,\!</math>
| J
| [M] [L]<sup>2</sup> [T]<sup>−2</sup>
|-valign="top"
! Total [[Energy operator|energy]]
|N/A
|Time-dependent potential:<br />
<math> \hat{E} = i \hbar \frac{\partial }{\partial t} \,\!</math>
 
Time-independent:<br />
<math> \hat{E} = E \,\!</math>
| J
| [M] [L]<sup>2</sup> [T]<sup>−2</sup>
|-valign="top"
! [[Hamiltonian operator|Hamiltonian]]
|
|<math> \begin{align} \hat{H} & = \hat{T} + \hat{V} \\
& = \frac{\bold{\hat{p}}\cdot\bold{\hat{p}}}{2m} + V \\
& = \frac{\hat{p}^2}{2m} + V \\
\end{align} \,\!</math>
| J
| [M] [L]<sup>2</sup> [T]<sup>−2</sup>
|-valign="top"
! [[Angular momentum operator]]
|<math>\begin{align}
\hat{L}_x & = -i\hbar \left(y {\partial\over \partial z} - z {\partial\over \partial y}\right)\\
\hat{L}_y & = -i\hbar \left(z {\partial\over \partial x} - x {\partial\over \partial z}\right)\\
\hat{L}_z & = -i\hbar \left(x {\partial\over \partial y} - y {\partial\over \partial x}\right)
\end{align}</math>
||<math>\mathbf{\hat{L}} = -i\hbar \mathbf{r} \times \nabla </math>
|| J s = N s m<sup>−1</sup>
|| [M] [L]<sup>2</sup> [T]<sup>−1</sup>
|-valign="top"
! [[Spin (physics)|Spin]] angular momentum
|<math>\begin{align}
\hat{S}_x & = {\hbar \over 2} \sigma_x \\
\hat{S}_y = {\hbar \over 2} \sigma_y \\
\hat{S}_z = {\hbar \over 2} \sigma_z
\end{align}</math>
 
where
 
<math>
\sigma_x = \begin{pmatrix}
0&1\\
1&0
\end{pmatrix}
</math>
 
<math>
\sigma_y = \begin{pmatrix}
0&-i\\
i&0
\end{pmatrix}
</math>
 
<math>
\sigma_z = \begin{pmatrix}
1&0\\
0&-1
\end{pmatrix}
</math>
 
are the [[pauli matrices]] for [[spin-½]] particles.
|<math>\mathbf{\hat{S}} = {\hbar \over 2} \boldsymbol{\sigma} \,\!</math>
 
where '''σ''' is the vector whose components are the pauli matrices.
| J s = N s m<sup>−1</sup>
| [M] [L]<sup>2</sup> [T]<sup>−1</sup>
|-valign="top"
 
! Total angular momentum
||<math>\begin{align}
\hat{J}_x & = \hat{L}_x + \hat{S}_x\\
\hat{J}_y & = \hat{L}_y + \hat{S}_y\\
\hat{J}_z & = \hat{L}_z + \hat{S}_z
\end{align}</math>
||<math>\begin{align}
\mathbf{\hat{J}} & = \mathbf{\hat{L}}+\mathbf{\hat{S}} \\
& = -i\hbar \bold{r}\times\nabla + \frac{\hbar}{2}\boldsymbol{\sigma}
\end{align}</math>
|| C m
|| [I] [T] [L]
|-valign="top"
! [[Transition dipole moment]] (electric)
||<math>\begin{align}
\hat{d}_x & = q\hat{x}\\
\hat{d}_y & = q\hat{y}\\
\hat{d}_z & = q\hat{z}
\end{align}</math>
||<math>\mathbf{\hat{d}} = q \mathbf{\hat{r}} </math>
|| C m
|| [I] [T] [L]
|-valign="top"
|}
 
===Examples of applying quantum operators===
 
The procedure for extracting information from a wave function is as follows. Consider the momentum ''p'' of a particle as an example. The momentum operator in one dimension is:
 
:<math>\hat{p} = -i\hbar\frac{\partial }{\partial x}</math>
 
Letting this act on ''ψ'' we obtain:
 
:<math>\hat{p} \psi = -i\hbar\frac{\partial }{\partial x} \psi ,</math>
 
if ''ψ'' is an eigenfunction of <math>\hat{p}</math>, then the momentum eigenvalue ''p'' is the value of the particle's momentum, found by:
 
:<math> -i\hbar\frac{\partial }{\partial x} \psi = p \psi.</math>
 
For three dimensions the momentum operator uses the [[nabla symbol|nabla]] operator to become:
 
:<math>\mathbf{\hat{p}} = -i\hbar\nabla .</math>
 
In Cartesian coordinates (using the standard Cartesian basis vectors '''e'''<sub>x</sub>, '''e'''<sub>y</sub>, '''e'''<sub>z</sub>) this can be written;
 
:<math>\mathbf{e}_\mathrm{x}\hat{p}_x + \mathbf{e}_\mathrm{y}\hat{p}_y + \mathbf{e}_\mathrm{z}\hat{p}_z = -i\hbar\left ( \mathbf{e}_\mathrm{x} \frac{\partial }{\partial x} + \mathbf{e}_\mathrm{y} \frac{\partial }{\partial y} + \mathbf{e}_\mathrm{z} \frac{\partial }{\partial z} \right ),</math>
 
that is:
 
:<math> \hat{p}_x = -i\hbar \frac{\partial}{\partial x}, \quad \hat{p}_y = -i\hbar \frac{\partial}{\partial y} , \quad \hat{p}_z = -i\hbar \frac{\partial}{\partial z} \,\!</math>
 
The process of finding eigenvalues is the same. Since this is a vector and operator equation, if ''ψ'' is an eigenfunction, then each component of the momentum operator will have an eigenvalue corresponding to that component of momentum. Acting <math> \mathbf{\hat{p}} </math> on ''ψ'' obtains:
 
:<math> \begin{align}
\hat{p}_x \psi & = -i\hbar \frac{\partial}{\partial x} \psi = p_x \psi \\
\hat{p}_y \psi & = -i\hbar \frac{\partial}{\partial y} \psi = p_y \psi \\
\hat{p}_z \psi & = -i\hbar \frac{\partial}{\partial z} \psi = p_z \psi \\
\end{align} \,\!</math>
 
==See also==
<div class="references-small" style="-moz-column-count:3; column-count:3;">
*[[Bounded linear operator]]
*[[Representation theory]]
</div>


==References==
==References==
* {{cite web
| first = Greg | last = Rose
| title = A precis of the new attacks on GSM encryption | publisher = [[QUALCOMM]] Australia | date = 2003-09-10
| url = http://www.qualcomm.com.au/PublicationsDocs/GSM_Attacks.pdf
}}
* {{cite journal
| first = Alexander | last =  Maximov |author2= Thomas Johansson|author3= Steve Babbage
| title = An Improved Correlation Attack on A5/1
| journal =  [[Selected Areas in Cryptography]] 2004 | year = 2004 | pages = 1–18
}}


{{reflist}}
==External links==
 
* {{cite web
{{Physics operator}}
| first = Marc | last = Briceno |author2= Ian Goldberg|author3= David Wagner | date = 1999-10-23
 
| url = http://edipermadi.files.wordpress.com/2008/03/pedagogical_implementation_of_a5_cipher.pdf
{{DEFAULTSORT:Operator (Physics)}}
| title =  A pedagogical implementation of the GSM A5/1 and A5/2 "voice privacy" encryption algorithms
[[Category:Operator theory]]
}}
[[Category:Theoretical physics]]
* {{cite web
| date =  2009-08-25
| url = http://www.neowin.net/news/main/09/08/25/huge-gsm-flaw-allows-hackers-to-listen-in-on-voice-calls
| title =  Huge GSM flaw allows hackers to listen in on voice calls
}}
* {{cite news
| url = http://www.cs.technion.ac.il/%7Ebarkan/GSM-Media/HaaretzInternetEnglish.pdf
| title = Technion team cracks GSM cellular phone encryption | work = [[Haaretz]]
| first = Hadar | last = Horesh | date = 2003-09-03
}}
* {{cite web
| url = http://www.cs.technion.ac.il/users/wwwb/cgi-bin/tr-info.cgi?2006/CS/CS-2006-07
| title = Instant Ciphertext-Only Cryptanalysis of GSM Encrypted Communication (Technical Report CS-2006-07)
| first = Elad | last =  Barkan |author2= Eli Biham|author3= Nathan Keller |date=July 2006
}}
* {{cite web
| url = http://www.ma.huji.ac.il/~nkeller
| title = Nathan Keller's Homepage
}}
* {{cite web
| url = http://l-system.net.pl/crypto/A5_1_stream_cipher.svg
| title = Animated SVG showing A5/1 stream cypher
}}
* {{cite web
| url = http://ipsec.pl/files/ipsec/security_in_the_gsm_network.pdf
| title = Security in the GSM network
| first = Marcin | last =  Olawski | date = May 2011 | publisher = IPSec.pl
}}
{{Cryptography navbox | stream}}


[[ar:مؤثر (فيزياء)]]
{{DEFAULTSORT:A5 1}}
[[de:Operator (Mathematik)#Operatoren der Physik]]
[[Category:Stream ciphers]]
[[fr:Opérateur (physique)]]
[[Category:Broken stream ciphers]]
[[lt:Operatoriai kvantinėje mechanikoje]]
[[Category:Mobile telecommunications standards]]
[[pt:Operador (física)]]
[[Category:3GPP standards]]
[[ru:Оператор (физика)]]
[[Category:GSM standard]]
[[zh:算符 (物理學)]]

Revision as of 07:29, 11 August 2014

A5/1 is a stream cipher used to provide over-the-air communication privacy in the GSM cellular telephone standard. It is one of seven algorithms which were specified for GSM use.[1] It was initially kept secret, but became public knowledge through leaks and reverse engineering. A number of serious weaknesses in the cipher have been identified.

History and usage

A5/1 is used in Europe and the United States. A5/2 was a deliberate weakening of the algorithm for certain export regions.[2] A5/1 was developed in 1987, when GSM was not yet considered for use outside Europe, and A5/2 was developed in 1989. Though both were initially kept secret, the general design was leaked in 1994 and the algorithms were entirely reverse engineered in 1999 by Marc Briceno from a GSM telephone. In 2000, around 130 million GSM customers relied on A5/1 to protect the confidentiality of their voice communications; by 2011, it was 4 billion.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park..

Security researcher Ross Anderson reported in 1994 that "there was a terrific row between the NATO signal intelligence agencies in the mid-1980s over whether GSM encryption should be strong or not. The Germans said it should be, as they shared a long border with the Warsaw Pact; but the other countries didn't feel this way, and the algorithm as now fielded is a French design."[3]

Description

The A5/1 stream cipher uses three LFSRs. A register is clocked if its clocking bit (orange) agrees with one or both of the clocking bits of the other two registers.

A GSM transmission is organised as sequences of bursts. In a typical channel and in one direction, one burst is sent every 4.615 milliseconds and contains 114 bits available for information. A5/1 is used to produce for each burst a 114 bit sequence of keystream which is XORed with the 114 bits prior to modulation. A5/1 is initialised using a 64-bit key together with a publicly known 22-bit frame number. Older fielded GSM implementations using Comp128v1 for key generation, had 10 of the key bits fixed at zero, resulting in an effective key length of 54 bits. This weakness was rectified with the introduction of Comp128v2 which yields proper 64 bits keys. When operating in GPRS / EDGE mode, higher bandwidth radio modulation allows for larger 348 bits frames, and A5/3 is then used in a stream cipher mode to maintain confidentiality.

A5/1 is based around a combination of three linear feedback shift registers (LFSRs) with irregular clocking. The three shift registers are specified as follows:

LFSR
number 		Length in
bits 		Feedback
polynomial 		Clocking
bit 		Tapped
bits
1 19 8 13, 16, 17, 18
2 22 10 20, 21
3 23 10 7, 20, 21, 22

The bits are indexed with the least significant bit (LSB) as 0.

The registers are clocked in a stop/go fashion using a majority rule. Each register has an associated clocking bit. At each cycle, the clocking bit of all three registers is examined and the majority bit is determined. A register is clocked if the clocking bit agrees with the majority bit. Hence at each step at least two or three registers are clocked, and each register steps with probability 3/4.

Initially, the registers are set to zero. Then for 64 cycles, the 64-bit secret key is mixed in according to the following scheme: in cycle , the ith key bit is added to the least significant bit of each register using XOR —

Each register is then clocked.

Similarly, the 22-bits of the frame number are added in 22 cycles. Then the entire system is clocked using the normal majority clocking mechanism for 100 cycles, with the output discarded. After this is completed, the cipher is ready to produce two 114 bit sequences of output keystream, first 114 for downlink, last 114 for uplink.

Security

The message on the screen of a mobile phone with the warning about lack of ciphering

A number of attacks on A5/1 have been published, and the American National Security Agency is able to routinely decrypt A5/1 messages according to released internal documents.[4]

Some attacks require an expensive preprocessing stage after which the cipher can be broken in minutes or seconds. Until recently, the weaknesses have been passive attacks using the known plaintext assumption. In 2003, more serious weaknesses were identified which can be exploited in the ciphertext-only scenario, or by an active attacker. In 2006 Elad Barkan, Eli Biham and Nathan Keller demonstrated attacks against A5/1, A5/3, or even GPRS that allow attackers to tap GSM mobile phone conversations and decrypt them either in real-time, or at any later time.

According to professor Jan Arild Audestad, at the standardization process which started in 1982, A5/1 was originally proposed to have a key length of 128 bits. At that time, 128 bits was projected to be secure for at least 15 years. It is now estimated that 128 bits would in fact also still be secure as of 2014. Audestad, Peter van der Arend, and Thomas Haug says that the British insisted on weaker encryption, with Haug saying he was told by the British delegate that this was to allow the British secret service to eavesdrop more easily. The British proposed a key length of 48 bits, while the West Germans wanted stronger encryption to protect against East German spying, so the compromise became a key length of 56 bits.[5]

Known-plaintext attacks

The first attack on the A5/1 was proposed by Ross Anderson in 1994. Anderson’s basic idea was to guess the complete content of the registers R1 and R2 and about half of the register R3. In this way the clocking of all three registers is determined and the second half of R3 can be computed.[3]

In 1997, Golic presented an attack based on solving sets of linear equations which has a time complexity of 240.16 (the units are in terms of number of solutions of a system of linear equations which are required).

In 2000, Alex Biryukov, Adi Shamir and David Wagner showed that A5/1 can be cryptanalysed in real time using a time-memory tradeoff attack,[6] based on earlier work by Jovan Golic.[7] One tradeoff allows an attacker to reconstruct the key in one second from two minutes of known plaintext or in several minutes from two seconds of known plain text, but he must first complete an expensive preprocessing stage which requires 248 steps to compute around 300 GB of data. Several tradeoffs between preprocessing, data requirements, attack time and memory complexity are possible.

The same year, Eli Biham and Orr Dunkelman also published an attack on A5/1 with a total work complexity of 239.91 A5/1 clockings given 220.8 bits of known plaintext. The attack requires 32 GB of data storage after a precomputation stage of 238.[8]

Ekdahl and Johannson published an attack on the initialisation procedure which breaks A5/1 in a few minutes using two to five minutes of conversation plaintext.[9] This attack does not require a preprocessing stage. In 2004, Maximov et al. improved this result to an attack requiring "less than one minute of computations, and a few seconds of known conversation". The attack was further improved by Elad Barkan and Eli Biham in 2005.[10]

Attacks on A5/1 as used in GSM

In 2003, Barkan et al. published several attacks on GSM encryption.[11] The first is an active attack. GSM phones can be convinced to use the much weaker A5/2 cipher briefly. A5/2 can be broken easily, and the phone uses the same key as for the stronger A5/1 algorithm. A second attack on A5/1 is outlined, a ciphertext-only time-memory tradeoff attack which requires a large amount of precomputation.

In 2006, Elad Barkan, Eli Biham, Nathan Keller published the full version of their 2003 paper, with attacks against A5/X Ciphers. The authors claim:[12] 36 year-old Diving Instructor (Open water ) Vancamp from Kuujjuaq, spends time with pursuits for instance gardening, public listed property developers in singapore developers in singapore and cigar smoking. Of late took some time to go China Danxia.

In 2007 Universities of Bochum and Kiel started a research project to create a massively parallel FPGA-based cryptographic accelerator COPACOBANA. COPACOBANA was the first commercially available solution[13] using fast time-memory trade-off techniques that could be used to attack the popular A5/1 and A5/2 algorithms, used in GSM voice encryption, as well as the Data Encryption Standard (DES). It also enables brute force attacks against GSM eliminating the need of large precomputated lookup tables.

In 2008, the group The Hackers Choice launched a project to develop a practical attack on A5/1. The attack requires the construction of a large look-up table of approximately 3 terabytes. Together with the scanning capabilities developed as part of the sister project, the group expected to be able to record any GSM call or SMS encrypted with A5/1, and within about 3–5 minutes derive the encryption key and hence listen to the call and read the SMS in clear. But the tables weren't released.[14]

A similar effort, the A5/1 Cracking Project, was announced at the 2009 Black Hat security conference by cryptographers Karsten Nohl and Sascha Krißler. It created the look-up tables using Nvidia GPGPUs via a peer-to-peer distributed computing architecture. Starting in the middle of September 2009, the project ran the equivalent of 12 Nvidia GeForce GTX 260. According to the authors, the approach can be used on any cipher with key size up to 64-bits.[15]

In December 2009, the A5/1 Cracking Project attack tables for A5/1 were announced by Chris Paget and Karsten Nohl. The tables use a combination of compression techniques, including rainbow tables and distinguished point chains. These tables constituted only parts of the 2TB completed table, and had been computed during three months using 40 distributed CUDA nodes, and then published over BitTorrent.[14][15][16][17] More recently the project has announced a switch to faster ATI Evergreen code, together with a change in the format of the tables and Frank A. Stevenson announced breaks of A5/1 using the ATI generated tables.

Documents leaked by Edward Snowden in 2013 states that NSA "can process encrypted A5/1".[18]

See also

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

  • Template:Cite web
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

External links

Template:Cryptography navbox

  1. ↑ Template:Cite web
  2. ↑ Template:Cite web
  3. ↑ 3.0 3.1 Template:Cite newsgroup
  4. ↑ NSA Able To Crack A5/1 Cellphone Crypto - Slashdot
  5. ↑ http://www.aftenposten.no/nyheter/uriks/Sources-We-were-pressured-to-weaken-the-mobile-security-in-the-80s-7413285.html#.UtBeNpD_sQs
  6. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  7. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  8. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  9. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  10. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  11. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  12. ↑ Template:Cite web
  13. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  14. ↑ 14.0 14.1 55 years old Systems Administrator Antony from Clarence Creek, really loves learning, PC Software and aerobics. Likes to travel and was inspired after making a journey to Historic Ensemble of the Potala Palace.

    You can view that web-site... ccleaner free download
  15. ↑ 15.0 15.1 https://har2009.org/program/attachments/119_GSM.A51.Cracking.Nohl.pdf Subverting the security base of GSM. Karsten Nohl and Sascha Krißler
  16. ↑ Template:Cite news
  17. ↑ Template:Cite news
  18. ↑ http://www.washingtonpost.com/business/technology/by-cracking-cellphone-code-nsa-has-capacity-for-decoding-private-conversations/2013/12/13/e119b598-612f-11e3-bf45-61f69f54fc5f_story.html
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