quantum_cryptography.htm

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explanation is in, let us say, parallel processing that is even more parallel
than any other that exists now. Yeah, it sounds stupid, and maybe I'm stupid,
but I can't put the idea of quantum computing in one sentence. </p>

<p align="left">&nbsp; </p>

<h4 align="left">6) The last (exit).... but not least </h4>

<p>&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp; Well, as we could see, the public crypto-systems like RSA
could become useless in the future (don't worry, you'll be grandpas by that
time), with the appearances of the first useable quantum computers (boxes in
further text). You have one possible way, and yet the simplest one, to secretly
send a message. You can always turn to secret-key systems, such as Vernam's
system described above, if you have the way to perfectly hide the secret key
from Eve. This is the exactly the moment where quantum physics enters the
scene.. Bob and Alice must share a secret key (in opposite of public-key
systems), and quantum cryptography allows two physically separated parties to
create random secret key without resorting to the services of courier. What's
even better it also allows them to verify that the key has not been intercepted.
Quantum cryptography is not therefore a totally new crypto-system, but the
procedure to distribute the key in perfect secrecy from other parties like Eve (hehehehe).
So, let me put this I've just said in few words; quantum crypto is not crypto
algorithm, but it allow a key to be securely distributed and is consequently a
natural complement to Vernam's cipher.</p>
<p align="center"><img border="0" src="QuantumCryptoS.jpg"></p>
<p align="center">&nbsp;&nbsp;&nbsp; To understand how quantum cryptography works we can
consider the &quot;BB84&quot; communication protocol, which was introduced in
1984 by Charles Bennett of IBM and Gilles Brassard from the University of
Montreal. Alice and Bob are connected by a quantum channel and a classical
public channel (see the picture above). If single photons are used to carry
information the quantum channel is usually optical fibre. The public channel,
however, can be any communication link, such as phone line or internet. Let us
stop now a little and say something about information. The information in
computer world is represented by series of 0's and 1's that assembled together
in defined order present information. That information can be anything numbers,
words, pictures, we only need to know how to interpret that binary information
(binary stands for there is information represented by series of 0 and 1, but
this is really out of topic so.... sorry I wasted eyes to those that well know
that, but I felt like saying it). Well, that 0 and 1 while traveling your phone
lines is represented like some voltage. Usually&nbsp; in the world of digital
electronics logical 0 and 1 are represented like 0V and 5V considering the
ground (sometimes -5V and 5V, and 0V can represent some other state). In the
case of quantum channel carriers are photons and as we could see we can use
polarization and phase shift.... can you dig it? Yes, we can define some
arbitrary angles of polarization or phase shift (well do you see now why was
that story so long). In practice, the public link is also optical fibre, with
both channels differing only in the intensity of light pulses. How this thing
work?</p>
<p align="center">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</p>
<p align="center">&nbsp;<img border="0" src="CrptoProcess.jpg" width="488" height="254"></p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;</p>
<p>1)&nbsp;&nbsp;&nbsp; First, Alice has four polarizers, which can transmit,
which can transmit single photons polarized either vertically, horizontally, at
+45 degrees, or at -45 degrees (look at the picture above). She sends a series
of photons down the quantum channel, having chosen at random one of the
polarization states for each photon (which in this case represents key, also
note on pictures which polarization angles represent 0's and 1's, IT'S NOT
MISTAKE, and IT'S VERY IMPORTANT to note so you could understand process).</p>
<p>2)&nbsp;&nbsp;&nbsp; Second, Bob has four analyzers, not two (that are
devices that can analyze polarization angle, or several angles at one time, but
note, when you put analyzer and there is single photon, if you set analyzer at
some angle(s) you will be able only to see if the photon is polarized at that
angle(s) or not, also when you perform measure once you can't measure
polarization of the same photon again with other analyzer set to measure some
other angle(s), because once you measure, the information is destroyed by
measuring because of interaction of the measuring system and system you measure,
this is represented in physics by so called projection theorem, because you
project the measured system onto system that measures, huh sounds confusing..
yeah I've told you that about quantum mechanics, in other words this all means
you can't measure ALL angles at once). One analyzer allows Bob to distinguish
between photons polarized at +45 degrees and -45 degrees (see the picture
above), and other allows him to distinguish between horizontally and vertically
polarized photons. Note that Alice has four polarizers and Bob has only two
analyzers! Also note how are 0's and 1's set by Alice. That is crucial! What Bob
does then? Well, he randomly chose one analyzer and uses it to record each
photon. He now writes down which analyzer he used and what it recorded (if he
chose wrong analyzer, he won't get any information about the state of photon, in
case you forgot, polarization is one of states of photon). For example, if Alice
sent a vertically polarized photon and Bob chose to detect photons at +/-45
degrees. Basically if Bob chose +/-45 degrees analyzer there is 50% chance that
he will record something, elementary my dear Watson. Even if Bob finds out later
that he chose the wrong analyzer, he will have no way of finding out which
polarization state Alice sent.</p>
<p>3)&nbsp;&nbsp;&nbsp; Third, after exchanging enough photons, Bob announces on
the public channel the sequence of analyzers he used, <b>but not the results he
obtained</b>.</p>
<p>4)&nbsp;&nbsp;&nbsp; Fourth, Alice compares this sequence with the list of
bits she originally sent, and tells Bob on the public channel on which occasions
his analyzer was compatible with the photon's polarization. She <b>does not tell
him which polarization states she sent</b>. If Bob used an analyzer that was not
compatible with Alice's photon, bit is simply discarded. For the bits that
remain (look at the retained bit sequence at picture), Alice an Bob know that
they have the same values - provided that the eavesdropper didn't perturb the
transmission. The <b>bits that are left</b> Alice and Bob can use to <b>generate
key</b> that they will use to encrypt the message they will send then by public
channel. </p>

<p align="center"><img border="0" src="QuantumCrypto.jpg"> </p>

<p>&nbsp;&nbsp;&nbsp; Let us now see the case when there is Eve. Suppose the Eve
has intercepted both quantum and Public channel (suppose Eve cut the fibre and
she set her equipment), and of course, she sends information to Bob so her
eavesdropping couldn't be noticed. What's wrong with that picture in this case?
Obviously, the disclosed bits cannot be used for encryption anymore. If Eve
intercepted their key, the correlation between the values of their bits will
have been reduced. For example if Eve had the same equipment like Bob and cuts
fibre and measures signal, she will always get random bit whenever she chooses
wrong analyzers (that is statistically 50% of all cases). But having intercepted
the signal Eve still has to send a photon to Bob to cover her tracks. Therefore,
in 50% of cases Alice's and Bob's analyzers match, but what's in case that Eve
didn't used a right analyzer and that is in 50% of cases? However in half of
these cases photon will accidentally pass through the right analyzer at Bob's
side. We can see now that correlation between Alice's and Bob's measures will
drop to only 25% in presence of Eve. In that case Alice and Bob will know that
information has been intercepted, when they compare keys over public link they
will see a greater disagreement (to be more precise, twice greater) and they
will drop transmission. Simple isn't it? </p>

<p>&nbsp; </p>

<h4><b>7) Quantum cryptography in real life</b> </h4>

<p>&nbsp; </p>

<p>&nbsp;&nbsp;&nbsp; So how you can achieve quantum cryptography in practice?
Photons are good candidates to carry information, they are easy to produce and
to measure. Story I've presented for polarization can be used same for phase
shift. In fact, it's more used than polarization. What's even better they can be
transmitted through the optical fibre and over last 25 years attenuation of
light (measure of how much photons are lost during transmission) at wavelenght
of 1300nm has been reduced from several decibels per metre to just 0.35 decibels
per kilometre. This means that photons can travel up to 10km before 50% of them
are lost which is sufficient to perform quantum cryptography in local networks.
Some of you with more technical education could note that you could use an
amplifier to transmit photons, but amplifiers cannot be used because quantum
states cannot be copies (in some cases yes, in case of quantum teleportation,
but this is not that case). There are also some projects aiming to establish
quantum communication from a satellite down to earth or other satellite, but as
far as I know this is not yet practice. </p>

<p>&nbsp;&nbsp;&nbsp; Of course, this is not only problem. There is always
trouble with quality of link. Uncorrelated bits may originate from several
experimental imperfections. First, Alice has to ensure that she creates photons
that are exactly the states she choose. If, for instance, a vertical photon is
incorrectly polarized at an angle 84 degrees, there is only 1% possibility that
Bob will find in channel for horizontally polarized photons. Similar problem is
from Bob's side; does he measure exactly 90 degrees. Another difficulty is
ensuring that the encoded bits are maintained during transmission. There is also
one more problem, due to the birefringence of the fibre, the polarization states
received by Bob will, in general, be different from those sent by Alice, and
that also asks for calibration of their apparatus, etc. etc. </p>

<p>&nbsp;&nbsp;&nbsp; To overcome these problems, Alice and Bob have to apply a
classical error-correction algorithm to their data so that they can reduce the
errors below an error rate of 10^(-9) (0.000000001 or one in billion) - the
industry standard for digital telecommunications. And since they cannot be sure
if the presence of of uncorrelated bits was due to the poor performance of their
set-up or to an eavesdropper, they have to assume the worst-case scenario - that
all the errors were caused by Eve. There is one procedure Alice and Bob may use
known as &quot;privacy amplification&quot; in which several bits are combined
into one. This procedure ensures that the combined bits correlate only if Alice
and Bob's initial bits are the same. The problem with privacy amplification is
that it shortens the key length a lot and it's only possible up to certain
error. That means that Alice and Bob have to be careful to introduce as few
errors as possible when they initially send their quantum bits. </p>

<h4>&nbsp; </h4>

<h4>8) Last words </h4>

<p>&nbsp; </p>

<p>&nbsp;&nbsp;&nbsp; In tradition of a dying tutorial I must now say something
to close this text. Well you've now been introduced to one of new technologies
that are now used. Don get caught in web... Port 80 is not only thing in
communications today, like the phone lines also aren't all. Optical communication
has been used for some 20 years, and today not only for T1 backbones. There are
some other technologies that go parallel with quantum cryptography, like quantum
computing and quantum teleportation that I will also present to you soon. I hope
that you've liked this tutorial and I hope I've make it readable. Don't be lazy
read it few times if you don't catch me (no, not because it's my tutorial)
because it's confusing and hard topic. I'll appreciate any comments and suggestions,
and feel free to ask me any question if you have it about quantum technology.
You have my mail. Thank you for your time... </p>

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