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AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION A BRIEF REVIEW THE NEXT GENERATION OF COMPUTING DEVICES? Meysam Madani Sharif University of Technology

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MEYSAM MADANI- MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 2 Quantum Mechanic A ReviewQuantum Computation Quantum Algorithms and its Implementations Quantum CryptographyQuantum Information

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MEYSAM MADANI- MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 3 Quantum Mechanic ReviewQuantum Computation Quantum Algorithms and Implementations Quantum CryptographyQuantum Information

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QUANTUM MECHANICS Quantum mechanics is the body of scientific principles that explains the behaviour of matter and its interactions with energy on the scale of atoms and subatomic particles. Classical physics explains matter and energy at the macroscopic level of the scale familiar to human experience On the other hand, at the end of the 19th century scientists discovered phenomena in both the large (macro) and the small (micro) worlds that classical physics could not explain. Coming to terms with these limitations led to the development of quantum mechanics, a major revolution in physics. MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 4

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QUANTUM MECHANIC Ludwig Eduard Boltzmann suggested in 1877 that the energy levels of a physical system, such as a molecule, could be discrete. In 1900, the German physicist Max Planck reluctantly introduced the idea that energy is quantized in order to derive a formula for the observed frequency dependence of the energy emitted by a black body, called Planck's Law, 1.I(ν,T) is the energy per unit time, 2. h is the Planck constant; 3. c is the speed of light in a vacuum; 4. k is the Boltzmann constant; 5.ν is the frequency of the electromagnetic radiation MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 5 Ludwig Eduard Boltzmann Max Planck

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QUANTUM MECHANIC tefan Procopiu in 1911—1913, and subsequently Niels Bohr in 1913, calculate the magnetic moment of the electron, which was later called the "magneton"; Subsequently made possible for both the magnetic moments of the proton and the neutron. In 1905, Einstein explained the photoelectric effect by postulating that light, or more generally all electromagnetic radiation, can be divided into a finite number of "energy quanta" that are localized points in space. These energy quanta later came to be called " photons “ It effectively solved the problem of black body radiation attaining infinite energy, MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 6 Niels Bohr Albert Einstein

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QUANTUM MECHANIC In 1924, Louis de Broglie put forward his theory of matter waves by stating that particles can exhibit wave characteristics and vice versa. In 1925, when the German physicists Werner Heisenberg and Adam Jonathon Davis developed matrix mechanics Erwin Schrödinger invented wave mechanics and the non- relativistic Schrödinger equation as an approximation to the generalized case of de Broglie's theory. Schrödinger subsequently showed that the two approaches were equivalent. Heisenberg formulated his uncertainty principle in 1927, MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 7 Werner Heisenberg Erwin Schrödinger

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QUANTUM MECHANIC Paul Dirac began the process of unifying quantum mechanics with special relativity by proposing the Dirac equation for the electron. The Dirac equation achieves the relativistic description of the wavefunction of an electron that Schrödinger failed to obtain. It predicts electron spin and led Dirac to predict the existence of the positron. He also pioneered the use of operator theory, including the influential bra-ket notation. John von Neumann formulated the mathematical basis for quantum mechanics as the theory of linear operators on Hilbert spaces MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 8 Paul Dirac John von Neumann

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PHOTONS Albert suggested that quantization was not just a mathematical trick: the energy in a beam of light occurs in individual packets, which are now called photons. The energy of a single photon is given by its frequency multiplied by Planck's constant: For centuries, scientists had debated between two possible theories of light: was it a wave or did it instead comprise a stream of tiny particles? The photoelectric effect puzzling wave theory. MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 9

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THE PHOTOELECTRIC EFFECT In 1887 Heinrich Hertz observed that light can eject electrons from metal. In 1902 Philipp Lenard discovered that the maximum possible energy of an ejected electron is related to the frequency of the light, not to its intensity. MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 10

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QUANTUM MECHANICS MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 11

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HEISENBERG - QUANTUM MECHANICS MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 12 Position or velocity: ? =2 Position: 5 =?

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YOUNG DOUBLE SLIT INTERFERENCE MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 13

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YOUNG DOUBLE SLIT INTERFERENCE MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 14

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YOUNG DOUBLE SLIT INTERFERENCE MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 15

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PHOTON MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 16

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PHOTON MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 17

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PHOTON MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 18

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PHOTON MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 19

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PHOTON MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 20

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PHOTON MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 21 Input B B NOT Output B.NOT.B=Id

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MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 22 Quantum Mechanic ReviewQuantum Computation Quantum Algorithms and Implementations Quantum CryptographyQuantum Information

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MODELS OF COMPUTATION Richard Feynman was the first to suggest, in a talk in 1981, that quantum-mechanical systems might be more powerful than classical computers. Feynman asked what kind of computer could simulate physics and then argued that only a quantum computer could simulate quantum physics efficiently. He focused on quantum physics rather than classical physics because, as he colorfully put it, nature isn't classical, dammit, and if you want to make a simulation of nature, you'd better make it quantum mechanical Around the same time, in “Quantum mechanical models of Turing machines that dissipate no energy“ Paul Benio demonstrated that quantum-mechanical systems could model Turing machines. In other words, he proved that quantum computation is at least as powerful as classical computation. But is quantum computation more powerful than classical computation? MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 23

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MODELS OF COMPUTATION David Deutsch explored this question and more in his 1985 paper “Quantum theory, the Church Turing principle and the universal quantum computer“ First, he introduced quantum counterparts to both the Turing machine and the universal Turing machine. He then demonstrated that the universal quantum computer can do things that the universal Turing machine cannot, including 1.generate genuinely random numbers 2.perform some parallel calculations in a single register 3.Perfectly simulate physical systems with finite dimensional state spaces. In 1989, in “Quantum computational networks", Deutsch described a second model for quantum computation: quantum circuits. MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 24

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MODELS OF COMPUTATION Deutsch demonstrated that quantum gates can be combined to achieve quantum computation in the same way that Boolean gates can be combined to achieve classical computation. He then showed that quantum circuits can compute anything that the universal quantum computer can compute, and vice versa. Andrew Chi-Chih Yao picked up where Deutsch left off and addressed the complexity of quantum computation in his 1993 paper “Quantum circuit complexity“ Specifically, he showed that any function that can be computed in polynomial time by a quantum Turing machine can also be computed by a quantum circuit of polynomial size. This finding allowed researchers to focus on quantum circuits, which are easier than quantum Turing machines. MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 25 David Deutsch Andrew Chi-Chih

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MODELS OF COMPUTATION Also in 1993, Ethan Bernstein and Umesh Vazirani presented ”Quantum complexity theory", in which they described a universal quantum Turing machine that can eficiently simulate any quantum Turing machine. MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 26 Umesh Vazirani Ethan Bernstein Peter Shor

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MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 27 Quantum Mechanic ReviewQuantum Computation Quantum Algorithms and Implementations Quantum CryptographyQuantum Information

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QUANTUM GATES In 1995, a cluster of articles examined which sets of quantum gates are adequate for quantum computation so that is, which sets of gates are sufficient for creating any given quantum circuit. Adriano Barenco et al. “Elementary gates for quantum computation“, showed that any quantum circuit can be constructed using nothing more than quantum gates on one qubit and controlled exclusive-OR gates on two qubits. David DiVincenzo, “Two-bit gates are universal for quantum computation“ proved that two-qubit quantum gates are adequate. Adriano Barenco, David Deutsch, and Artur Ekert showed that quantum controlled-NOT gates and one-qubit gates are together adequate; MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 28 David DiVincenzo Artur Ekert

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QUANTUM ALGORITHMS AND IMPLEMENTATIONS In 1992, David Deutsch and Richard Jozsa coauthored “Rapid solution of problems by quantum computation". They presented an algorithm that determines whether a function f is constant over all inputs or balanced. The Deutsch-Jozsa algorithm was the first quantum algorithm to run faster than its classical counterparts. L. Chuang et al. detailed how they used bulk nuclear magnetic resonance techniques to implement a simplifed version of the Deutsch-Jozsa algorithm. In Quantum complexity theory“, Bernstein and Vazirani were the first to identify a problem that can be solved in polynomial time by a quantum algorithm but requires superpolynomial time classically. Daniel R. Simon introduced a problem that a quantum algorithm can solve exponentially faster than any known classical algorithm. MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 29 Richard Jozsa

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QUANTUM ALGORITHMS AND IMPLEMENTATIONS This research inspired Peter W. Shor, who then invented two quantum algorithms that outshone all others: polynomial-time algorithms for finding prime factors and discrete logarithms, problems widely believed to require exponential time on classical computers. Simon and Shor both presented their discoveries at the 1994, “Algorithms for quantum computation: Discrete logarithms and factoring“. Shor's factorization algorithm in particular heightened excitement and even generated anxiety about the power and promise of quantum computing. To pose a practical threat to RSA cryptography, Shor's algorithm must be implemented on quantum computers that can hold and manipulate large numbers, and these do not exist yet. Isaac L. Chuang and his research team made headlines when they factored the number 15 on a quantum computer with seven qubits. MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 30 Isaac L. Chuang

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BQP, BPP, PSPACE, P, NP AND …. MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 31 P NP PSPACE BQP BPP Factoring Problem Quantum Polynomial Algorithms

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QUANTUM ALGORITHMS AND IMPLEMENTATIONS Lov K. Grover's algorithm for searching an unordered list, described in both “A fast quantum mechanical algorithm for database search" and “Quantum mechanics helps in searching for a needle in a haystack“, Unlike Shor's algorithm, Grover's algorithm solves a problem for which there are polynomial-time classical algorithms; however, Grover's algorithm does it quadratically faster than classical algorithms can. In 2003, Peter W. Shor addressed this stagnation in a short article called “Why haven't more quantum algorithms been found?“Shor oered several possible explanations, 1.The possibility that computer scientists have not yet developed intuitions for quantum behavior. 2.The article should be required reading for all computer science students, whose intuitionsare still being formed. MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 32 Lov K. Grover

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CANDIDATES FOR QUANTUM COMPUTERS 1.Superconductor-based quantum computers (including SQUID-based quantum computers) 2.Ion trap-based quantum computers 3."Nuclear magnetic resonance on molecules in solution"-based 4.“Quantum dot on surface"-based 5.“Laser acting on floating ions (in vacuum)"-based (Ion trapping) 6."Cavity quantum electrodynamics" (CQED)-based 7.Molecular magnet-based 8.Fullerene-based ESR quantum computer 9.Solid state NMR Kane quantum computer MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 33

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MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 34

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QUANTUM COMPUTING PROBLEMS 1.Current technology ≈ 40 Qubit operating machine needed to rival current classical equivalents. 2. Errors Decoherence - the tendency of a quantum computer to decay from a given quantum state into an incoherent state as it interacts with the environment. Interactions are unavoidable and induce breakdown of information stored in the quantum computer resulting in computation errors. Error rates are typically proportional to the ratio of operating time to decoherence time operations must be completed much quicker than the decoherence time. MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 36

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MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 37 Quantum Mechanic ReviewQuantum Computation Quantum Algorithms and Implementations Quantum CryptographyQuantum Information

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QUANTUM CRYPTOGRAPHY Shor's factorization algorithm has yet to be implemented on more than a few qubits. But if the efficient factorization of large numbers becomes possible, RSA cryptography will need to be replaced by a new form of cryptography, The cryptographic method in question is quantum key distribution, which was introduced in 1984 by Charles H. Bennett and Gilles Brassard in “Quantum cryptography: Public key distribution and coin tossing“ called BB84. In short, quantum key distribution is secure not because messages are encrypted in some difficult-to-decrypt way but rather because eavesdroppers cannot intercept messages undetected, regardless of computational resources. MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 38 Gilles Brassard Charles H. Bennett

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QUANTUM CRYPTOGRAPHY Quantum cryptography describes the use of quantum mechanical effects (in particular quantum communication and quantum computation) to perform cryptographic tasks or to break cryptographic systems. quantum key distribution and (hypothetical) use of quantum computers, For example, quantum mechanics guarantees that measuring quantum data disturbs that data; this can be used to detect eavesdropping in quantum key distribution. 1.Position-based quantum cryptography 2.Quantum commitment 3.Quantum key distribution 4.Post-quantum cryptography MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 39

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POST-QUANTUM CRYPTOGRAPHY In a predictive sense, quantum computers may become a technological reality; it is therefore important to study cryptographic schemes that are (supposedly) secure even against adversaries with access to a quantum computer. The need for post-quantum cryptography arises from the fact that many popular encryption and signature schemes RSA and its variants, and schemes based on elliptic curves, can be broken using Shor's algorithm for factoring and computing discrete logarithms on a quantum computer.discrete logarithms Examples for schemes that are, as of today's knowledge, secure against quantum adversaries are 1. McEliece schemes 2. Lattice-based schemes MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 40

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QUANTUM KEY DISTRIBUTION The most well known and developed application of quantum cryptography is quantum key distribution (QKD) QKD describes the process of using quantum communication to establish a shared key between two parties (usually called Alice and Bob) without a third party (Eve) learning anything about that key, even if Eve can eavesdrop on all communication between Alice and Bob. This is achieved by Alice encoding the bits of the key as quantum data and sending them to Bob, If Eve tries to learn these bits, the messages will be disturbed and Alice and Bob will notice. The security of QKD can be proven mathematically without imposing any restrictions on the abilities of an eavesdropper, something not possible with classical key distribution. Eve should not be able to impersonate Alice or Bob as otherwise a man-in-the-middle attack would be possible. MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 41

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QUANTUM KEY DISTRIBUTION NETWORKS 1.In 2004, the world's first bank transfer using quantum key distribution was carried in Vienna, Austria. 2. Swiss company Id Quantique was used in the Swiss canton (state) of Geneva to transmit ballot results to the capitol in the national election occurring on October 21, The DARPA Quantum network, a 10-node quantum key distribution network, has been running since 2004 in Massachusetts, USA. 4.The world's first computer network protected by quantum key distribution was implemented in October 2008, at a scientific conference in Vienna. SECOQC (Secure Communication Based on Quantum Cryptography) used 200 km of standard fibre optic cable to interconnect six locations across Vienna and the town of St Poelten located 69 km to the west. 5.SwissQuantum network, installed in the Geneva metropolitan area in March 2009, was to validate the reliability and robustness of QKD in continuous operation over a long time period in a field environment. ran stably for nearly 2 years until the completion of the project in January 2011, confirming the viability of QKD as a commercial encryption technology. 6.Tokyo QKD Network, UQCC2010, the network involves an international collaboration between 7 partners: NEC, Mitsubishi Electric, NTT and NICT from Japan, Toshiba (UK), Id Quantique (Switzerland) MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 42

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QUANTUM COMMITMENT 1.Following the discovery of quantum key distribution and its unconditional security, researchers tried to achieve other cryptographic tasks with unconditional security. 2.A commitment scheme allows a party Alice to fix a certain value (to "commit") in such a way that Alice cannot change that value while at the same time ensuring that the recipient Bob cannot learn anything about that value until Alice decides to reveal it. 3.In the quantum setting, they would be particularly useful 4.Crépeau and Kilian showed that from a commitment and a quantum channel, one can construct an unconditionally secure protocol for performing so-called oblivious transfer. 5.Unfortunately, early quantum commitment protocols were shown to be flawed. 6.In fact, Mayers showed that (unconditionally secure) quantum commitment is impossible: a computationally unlimited attacker can break any quantum commitment protocol. MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 43

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POSITION-BASED QUANTUM CRYPTOGRAPHY 1.The goal of position-based quantum cryptography is to use the geographical location of a player as its (only) credential. 2. For example, one wants to send a message to a player at a specified position with the guarantee that it can only be read if the receiving party is located at that particular position. 3.It has been shown by Chandran et al. that position-verification using classical protocols is impossible against colluding adversaries(who control all positions except the prover's claimed position). 4.Under the name of 'quantum tagging', the first position-based quantum schemes have been investigated in 2002 by Kent. A US-patent. 5.After several other quantum protocols for position verification have been suggested in Buhrman et al. were able to show a general impossibility result:: using an enormous amount of quantum entanglement, colluding adversaries are always able to make it look to the verifiers as if they were at the claimed position. 7. However, this result does not exclude the possibility of practical schemes in the bounded- or noisy-quantum-storage model (see above). MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 44

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QUANTUM CRYPTOGRAPHY MANUFACTURERS MagiQ Technologies id Quantique (http://www.idquantique.com/), Smart Quantum (http://www.smartquantum.com/). MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 45

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STEMPUNK BB84 C. Bennett, F. Bessette, G. Brassard, L. Savail, J. Smolin J. Cryptology 5, 3 (1992) MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 46

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QUANTUM CRYPTOGRAPHY Larger distances (up to 144km demonstrated) to test for satellite – earth links Munich/Vienna/Bristol: T. Schmitt-Manderbach et al., Phys. Rev. Lett. 98, (2007) Larger key rates: VCSEL lasers, detectors with better timing resolution, high clock rate ~Mbit/sec key rate (detector limited) NIST Gaithersburg: J.C. Bienfang et al. Optics Express 12, 2011 (2004 MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 47

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MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 48 Quantum Mechanic ReviewQuantum Computation Quantum Algorithms and Implementations Quantum CryptographyQuantum Information

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QUANTUM INFORMATION Secure channels of communication are of course crucial, but security is not the only consideration in the transfer of information. Accordingly, quantum cryptography is just one of several topics in the burgeoning field of quantum information. Other topics include quantum error correction, fault-tolerant quantum computation, quantum data compression, and quantum teleportation. Information needs to be protected not just from eavesdroppers but also from errors caused by channel noise, implementation flaws, and, in the quantum case, decoherence. Peter W. Shor, a trailblazer not just of quantum algorithms but also of quantum error correction and fault-tolerant quantum computation, was the first to describe a quantum error-correcting method. In his 1995 article “Scheme for reducing decoherence in quantum computer memory“ he demonstrated that encoding each qubit of information into nine qubits could provide some protection against decoherence. MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 49

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QUANTUM INFORMATION At almost the same time but without knowledge of Shor's article, Andrew M. Steane wrote “Error correcting codes in quantum theory“which achieved similar results. Very shortly thereafter, Shor and A.R. Calderbank presented improved results in “Good quantum error-correcting codes exist“ Error is not the only thing information theorists strive to reduce; they also seek to reduce the space required to represent information. In this 1948 paper, Shannon showed that it is possible, up to a certain limit, to compress data without loss of information. Almost 50 years later, Benjamin Schumacher developed a quantum version of Shannon's theorem. MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 50 Andrew M. Steane Benjamin Schumacher

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QUANTUM INFORMATION Not everything in quantum information theory has a precedent in classical information theory. In 1993, Charles H. Bennett et al. dazzled the scientific c community and delighted science fiction fans by showing that quantum teleportation is theoretically possible. They described how an unknown quantum state could be disassembled and then reconstructed perfectly in another location. The first researchers to verify this method of teleportation experimentally were Dik Bouwmeester et al., who reported their achievement in 1997 in “Experimental quantum teleportation" MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 51 Dirk Bouwmeester Claude E. Shannon

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REFERENCES 1.Historical Bibliography of Quantum Computing, Jill Cirasella, Brooklyn College Library. 2.Timeline of quantum computing, Wikipedia, 3.Quantum cryptography, Wikipedia, 4.Quantum key distribution, Wikipedia, 5.MagiQ Technologies, 6.Practical Quantum cryptography and possible attacks, Alex Ling, et. Al, NUS,. MEYSAM MADANI - AN INTRODUCTION TO QUANTUM COMPUTING CRYPTOGRAPHY AND INFORMATION 52

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