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<h4><font size="5">Quantum Cryptography</font></h4>
<p>(by Caboom, 24.01.2000)</p>
<p> </p>
<h4>1) And God said let there be... tutorial</h4>
<p> </p>
<p> This essay can be little out of usual hacking/cracking topic, but I wanted to present you a new technologies that are now used.
Also, this tutorial has a 'growing intension' so it will be updated at time to time.
The needing for update is mainly here because of need for better explanation
because it's hard topic even to those that are familiar with quantum physics and
this tutorial is not written for people that are familiar with quantum physics
and will give you only basic idea about quantum cryptography. I would be very
thankful for any suggestion how to make this tutorial better and clearer because
it's very hard to explain some details of quantum theory to people that are not
familiar with it. I will be most satisfied if this tutorial can read even
my grandmother and get something from it.</p>
<p> </p>
<h4>2) Classical heaven => Quantum hell</h4>
<p>
</p>
<p> Ok, so let's get to basics of quantum theory. First, let
me explain the term '<i>quantum</i>'. As a more careful observer can notice the close
relative to the word quantum is a word quantity. At the very beginning of this
century physicians noticed that there is something wrong with the classical
theory, to be more precise, they've noticed that their perspective of the
fundamental particles they had are not matching the experimental results they
got. The logical conclusion was that there is something wrong with the classical
idea they had about atom. <i> Max Planck</i> therefore has introduced the new approach
to the matter. In the classical physics, electron orbiting (orbiting around is
also relative term, I'll explain it little bit later) around the core or the
atom could have any possible energy, and related to this, could orbit around the
core at any possible distance. The problem with that vision was that because the
electron is negative charged and core, because consisted of positive protons and
neutral neutrons, was positive charged it was expected that electron would collapse
into core for about 0.0000000001 sec (because of different charges, core
attracts electron). The answer was that electron can't have any energy while
orbiting around core, energy of electron is <i> quanted.</i> This means that electron
has strongly defined possible energies, and can orbit only on some distances
around atom. To be more precise, even that is not definitely true, the electron
is not orbiting exactly around orbits, orbits are only defining the place where
is best possibility to find electron. Confused? You should be... at least
shocked if you've never heard about it. This brings us to the rather confusing
and paradoxal world of
quantum mechanics (quantum mechanics - one of parts of physics dealing with the
phenomena of the small world, basically it research the movement of the
particles, parallel to ordinary mechanics in classical world, but much wider).
In world of quantum mechanic we are not talking about the exact value, but the
possible value. To make it more clear, say you want to find the place where is
the particle you're especially interested in, you won't be able to find
the exact spot an say here it is, what you'll get will be something like beaver
that had car accident... one relatively wide circle darkest in the center...
That center is the place where is the greatest probability to find the particle,
but not necessary the place where particle really is. That is consequence of our
impossibility to make perfect measurement. In our world of 'big' objects,
measurements we take are good enough to say 'beaver was 1.5m long' (hmmm... long
beaver), but if you look more careful 1.5m could be 1.485755432m or 1.49532221m
or 1.5000000m, do you get the point, what is behind? You always have error in
measurement. In the world of small object, measurements are so sensitive that
you can't for instance measure the position and impulse (impulse is
product of mass and speed, it determines the movement) of particle, that is for
one of fundamentals of quantum mechanic. Of great importance is to understand
that physics is based on measurement, not on the theory. So let us make
conclusion... Quantum mechanic is dealing with 'small' world of atoms and
fundamental particles, while classical physics founded mainly by Isaac Newton is
dealing with the 'big' world of beavers... In quantum world all observables
(things that we can measure) have discrete values (that means that you can't
have any value of observable, for instance, electron can't have any energy while
orbiting around atom), and we can speak only about probability for real
events... there are many more rules, but these are one of the basic. If you want
to know more, you can take some of many good quantum mechanics books, but
watch... be sure that you're good with linear algebra and you've been through
the higher courses of math, or avoid books with equations.
</p>
<p>
</p>
<h4>3) ...and God also said let there be light
</h4>
<p>
</p>
<p> We've now been through the basics of quantum mechanics,
don't give up... I have to explain to you some facts about light also before I
get to the point. The phenomena of matter is that it's constructed from smaller
parts. The smallest ones that we know that build all other bigger constructions
of nature, like atoms and molecules we call <i>fundamental particles</i>.
What physicians have found was that particles are not particles in a sense of
marbles that are wondering around, but they are also waves. Let me break one
more illusion for you. Particles are not looking like balls, as the matter of
fact you can't say that particle has any shape. If someone shows you a picture
of a black blurred ball, and tells you: "it's the picture of
electron", he is probably showing you a picture of energy distribution of
electron (... huh it doesn't matter, it's just not the picture of exact particle,
remember that with probability? It's that story). What I'm pointing at? In this
'quantum cryptography' play main role is played by light, so I have to say
something about light and terms like <i>polarization</i> and <i>phase
shift</i>. Let me explain that wave-particle thing. You've probably heard for
photons, and you've probably heard for electro-magnetic waves. See the usual
picture of electromagnetic wave (E - electrical field, B - magnetic field,
x-some direction the light is traveling)
</p>
<p>
</p>
<p align="center"> <img border="0" src="Electro.jpg">
</p>
<p>
</p>
<p> Well that are two views on the same thing we call light.
The light has really bad manners, I could say the light is acting little bit
schizophrenic. It determines of the measure we take, we can see that light is
acting like electro-magnetic wave, or that the light is made of particles we
call photons. No, it's not like water, where we have great amount of particles
that interact and make waves we can see. This is result of fundamental organization
of nature that we call duality. I will stop at this moment with further
explanations why is this that way. Yes there is deeper reason, but... let us
stop here ok?
</p>
<p> Let me now explain terms <i>polarization</i> and <i>phase
shift. </i>Do you see that x axis? It the one represented with the long black
line in the middle of graphic. Well, why couldn't you rotate other two axes
around it? There is not any reason why you couldn't rotate the whole picture
around that axis. Let us suppose we have two waves of same wavelength (oh yes,
this is the one more flavors of light, actually that is the value that determines
the energy of wave, and this value determines is this a radio wave, normal
visible light, x - ray, gamma ray etc.), and let them travel the same path x.
But what about that other two axes E and B? One wave must have axes E and B at
the constant angle 90 degrees (you'll also see in further text expressions like
Pi, Pi/2, that is the other way to mark the angles, Pi = 180 degrees, yes that
is the same one Pi = 3.14... Ok, now you know you can rotate those two axes E
and B around axis x and what you can do with it? If you for instance take some arbitrary
position of vectors E and B, let it be position of axes you can see in figure 1
on the picture bellow:
</p>
<p align="center"> <br>
<img border="0" src="Axes.jpg">
</p>
<p> </p>
<p> If you take positions of vectors E and B like on figure 1
as the arbitrary (polarization is not definitive term, there is not existing
absolute position of vectors E and B but you must first define some
arbitrary position of vectors E and B and then consider the term of polarization),
you can see if you look a little bit closer on figure 2 that vectors E and B are
rotated for 90 degrees anticlockwise (or, you can also say rotated for Pi/2). We
now can finally define the term of polarization. You can say that the
electro-magnetic wave on figure 2 is polarized for 90 degrees (of course,
compared to our arbitrary polarization on figure 1). Even one photon, if we
consider particle side of light, can be polarized, why? Because, as I've
explained before, the light is booth particle and wave, and even one photon can
be considered also for wave (sorry, not further explanations for now...), and
you can also say that photon is polarized at some angle. When you use some light
source like bulb you have electro-magnetic waves of all possible angles of
polarizations. You can get especially polarized light at some angle with
the piece of equipment called <i>polarizator</i>.</p>
<p> Now there is only one term left to define, <i>phase
shift. </i>You could see on the picture above (one 3D picture) that function
that describe electro-magnetic wave really looks like wave. That is Sine
function, and you can see that is periodical function. Now I will tell you that
period of this function is 360 degrees or 2Pi. That 'periodical' means that the
function is repeating it's image after some value x. Look now on the picture bellow:</p>
<p> </p>
<p align="center"><img border="0" src="Phase.jpg"></p>
<p> </p>
<p> You can see that the green and red functions are the same
one function, but shifted on the x axis. To be more precise i will say that
equation for red function is Sin(x) and equation for green function is Sin(x+Pi/4).
I can now say that green function is <i>phase shifted for Pi/4 (45 degrees)</i>
considering red function. Same thing is for electromagnetic wave because electromagnetic
wave is described with sine function (that is one of possible ways to describe
electromagnetic wave, but for purpose of this tutorial this is good enough).</p>
<p> </p>
<h4>4) To those that are still with us...</h4>
<p> </p>
<p> Now that are basic terms of physics needed for purpose of
further reading explained, we can now pass to the basic topic of this text, and
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