IntroductionQuantum MechanicsSpin and PolarizationQuantum Coding SchemeBB84 Quantum Key Distribution ProtocolPractical problems with Quantum CryptographyConclusionReferencesQuantum CryptographyBy Prafulla Basavaraja(CS 265 – Spring 2005)Quantum Cryptography is one of the latest methods of security in the cipher world andhas been proclaimed as the ultimate in security. In this paper we will try to gain anunderstanding of how quantum cryptography works and what makes it so secure.IntroductionAll of the current day classical computer cryptography are based on certain class ofmathematical operations that are easy to perform in one direction but are extremelydifficulty in the other direction. For example, the very popular public key cryptosystem,RSA, gets its security from the difficulty of factoring large numbers. As the number ofdigits increase the difficulty of factoring grows rapidly. For example, a 128 digit numberwould take millions of years to factor with current computers.The limitation of the classical, purely mathematical, theory of computation is highlightedby a key observation made by Rolf Landauer back in the 1960s that information cannotbe separated from its physical representation. In classical physics, information is storedand processed in binary form. In the 1980s, C.Bennet, P.Benioff, R.Feynman and othersobserved that new and very powerful ways of information processing are possible withquantum mechanical systems. This gave birth to the concept of quantum computing. Using new algorithms such as Shor's algorithm, a quantum computer can solve the primefactors of very large numbers in polynomial time which would otherwise take millions ofyears. If quantum computing becomes a reality then, the above mentioned RSA algorithmwill become insecure. Fortunately, there exists an unbreakable code. As per Shannon's rules, if the data being encrypted is either much longer than the key orif the key is used repeatedly then, with sufficient computation power, it is possible toinfer the message. So to have an unbreakable code, we need a key that is as long as thedata and that are not repeated. This is the Vernam code or one-time pad. The biggestpractical problem of such an encryption scheme is the difficulty of distributing the key. Ifthe key could be secretly distributed then the data itself could be distributed. The solutionto the problem of key distribution was provided by C.Bennett and G.Brassard who firstsuggested how to use quantum mechanics for secure key distribution.So in order to understand quantum cryptography, we will first take a look at quantummechanics. After which we will take a look at one of the protocols that exploits theproperties of quantum mechanics for quantum key distribution. Quantum signature isanother application of quantum cryptography. In this paper we shall not cover quantumsignature. However it must be mentioned that the birth of the idea of quantumcryptography began with Wiesner's PhD thesis idea (in 1960s) of use of quantum moneyas money that can not be counterfeited.Page 1 of 6Quantum MechanicsQuantum mechanics deals with the behavior of elementary particles, atoms, and energy interms of probabilities. It is developed from Planck’s quantum principle and Heisenberg’suncertainty principle. The principle idea is that energy, momentum, and angularmomentums as well as charges come in discrete amounts called quanta. This ideadescribes the nature of photons. Photons are discrete bundles of energy that make uplight. The rules that describe the behavior of quanta are quite different from the Newtonianrules that describe the behavior of objects that we see around us in our day-to-day life.Superposition, Entanglement and Measurement problem are some of the basic rules thatdescribe the behavior of quanta.From the theory of superposition principle we know that whenever two (or more) wavesare traveling the same medium at the same time then they pass through each otherwithout being disturbed and the net displacement of the medium at any point in space ortime is simply the sum of the individual wave displacements. When we applysuperposition principle to quantum mechanics it can be seen that the position or energy ofthe photon can simultaneously possess two or more values. It was observed in the doubleslit experiment that when a single photon emitted from a ray gun it formed aninterference pattern with itself when passing through the 2 slits. See figure 1. Throughthis we see that the photon seems to have traveled down both paths at the same time thusbeing in 2 places simultaneously.The entanglement property applies to a pair of quanta where each is described withreference to another even though they are spatially separated. Specifically, a pair ofentangled photons has opposite rotational directions or spin with the total ‘spin’ of thesystem being zero. The important implication of this property is that the measurement ofspin of one gives the spin of the other.[1][2]The measurement of any measurable property of a photon disturbs its state. This is themeasurement problem. However, this fact provides the advantage that the presence of aneavesdropper can be detected.Before we move on to how this is used in quantum cryptography there are a few moreaspects of physics of photons with which we need to familiarize ourselves.Spin and PolarizationWe have already stated that photons exhibit spin. Polarization is the spin propensity of aphoton. A photon has an electric and magnetic fields represented by vectors perpendicularboth to each other and the direction of travel. The behavior of the electric field vectordetermines the polarization of a photon.Page 2 of 6A Polaroid allows photons to pass through only if their electric field is oscillating in oneparticular direction. So with the use of Polaroid, we can obtain a beam of light with all ofits photons having the same polarization.Let us next explore a quantum coding scheme that makes use of the polarization propertyof photons.Quantum Coding SchemeAs described in the previous section, polarization and measurement of polarization ofphotons can be done with the use of Polaroid. For the purpose of evolving a simplecoding scheme, let us consider only the rectilinear and diagonal polarization schemes.This gives us 4 directions of polarization of a photon. See figure below.The 4
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