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What is cryptography? Cryptography (derived from the Greek words kryptos and grapheinmeaning hidden writing) is the science of codes and ciphers. .
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A cipher is essentially a cryptographic algorithm which is used to convert a message, known as the plaintext, into unreadable text.
Then the message can then be safely transmitted without fear of letting sensitive information fall into the hands of the enemy.
What is cipher?
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The process Plaintext Key Sender Encryption Cryptotext Secure transmission Decryption Recipient Plaintext Key ready for use Message encryption Secure key distribution Hard Problem for conventional encryption
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The classic cryptography Encryption algorithm and related key are kept secret. Breaking the system is hard due to large numbers of possible keys. For example: for a key 128 bits long there are keys to check using brute force. The fundamental difficulty is key distribution to parties who want to exchange messages.
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PKC :the modern cryptography In 1970s the Public Key Cryptography emerged. Each user has two mutually inverse keys, The encryption key is published; The decryption key is kept secret. Anybody can send a message to x but only xcan read it.
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RSA The most widely used PKC is the RSA algorithm based on the difficulty of factoring a product ot two large primes. Easy ProblemHard Problem Given n compute p and q. Given two large primes p and q compute
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Light waves are propagated as discrete quanta called photons. They are massless and have energy, momentum and angular momentum called spin. Spin carries the polarization. If on its way we put a polarization filter a photon may pass through it or may not. We can use a detector to check of a photon has passed through a filter. Elements of the Quantum Theory
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Introduction Spawned during the last century Describes properties and interaction between matter at small distance scales Quantum state determined by(among others) Positions Velocities Polarizations Spins qubits
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Binary information Each photon carries one qubitof information Polarization can be used to represent a 0 or 1. In quantum computation this is called qubit.
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Notation Bra/Ket notation (pronounced “bracket”) From Dirac 1958 Each state represented by a vector denoted by a arrow pointing in the direction of the polarization
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Notation Simplified Bra/Ket-notation in this presentation Representation of polarized photons: horizontally: vertically: diagonally: and
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Polarized photons Measurement of a state not only measures but actually transforms that state to one of the basis vectors and If we chose the basis vectors and when measuring the state of the photon, the result will tell us that the photon's polarization is either or , nothing in between. ψ b a
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Experiment Classical experiment Equipment: laser pointer three polarization filters The beam of light i pointed toward a screen. The three filters are polarized at , and respectively
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Experiment The filter is put in front of the screen Light on outgoing side of filter is now 50% of original intensity
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Experiment Next we insert a filter whereas no light continue on the output side
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Experiment Here is the puzzling part… We insert a filter in between This increases the number of photons passing through
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Experiment explained Filter is hit by photons in random states. It will measure half of the photons polarized as
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Experiment explained Filter is perpendicular to that and will measure the photons with respect to , which none of the incoming photons match
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Experiment explained Filter measures the state with respect to the basis {, }
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Experiment explained Photons reaching filter will be measured as with 50% chance. These photons will be measured by filter as with 50% probability and thereby 12,5% of the original light pass through all three filters.
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Key distribution Alice and Bob first agree on two representations for ones and zeroes One for each basis used, {,}and {, }. This agreement can be done in public Define1 = 0 = 1 = 0 =
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Key distribution - BB84 Alice sends a sequence of photons to Bob.Each photon in a state with polarization corresponding to 1 or 0, but with randomly chosen basis. Bob measures the state of the photons he receives, with each state measured with respect to randomly chosen basis. Alice and Bob communicates via an open channel. For each photon, they reveal which basis was used for encoding and decoding respectively. All photons which has been encoded and decoded with the same basis are kept, while all those where the basis don't agree are discarded.
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Eavesdropping Eve has to randomly select basis for her measurement Her basis will be wrong in 50% of the time. Whatever basis Eve chose she will measure 1 or 0 When Eve picks the wrong basis, there is 50% chance that she'll measure the right value of the bit E.g. Alice sends a photon with state corresponding to 1 in the {,} basis. Eve picks the {, } basis for her measurement which this time happens to give a 1 as result, which is correct.
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Eves problem Eve has to re-send all the photons to Bob Will introduce an error, since Eve don't know the correct basis used by Alice Bob will detect an increased error rate Still possible for Eve to eavesdrop just a few photons, and hope that this will not increase the error to an alarming rate. If so, Eve would have at least partial knowledge of the key.
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Detecting eavesdropping When Alice and Bob need to test for eavesdropping By randomly selecting a number of bits from the key and compute its error rate Error rate < Emax assume no eavesdropping Error rate > Emax assume eavesdropping(or the channel is unexpectedly noisy)Alice and Bob should then discard the whole key and start over
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Noise Noise might introduce errors A detector might detect a photon even though there are no photons Solution: send the photons according to a time schedule. then Bob knows when to expect a photon, and can discard those that doesn't fit into the scheme's time window. There also has to be some kind of error correction in the over all process.
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Error correction Suggested by Hoi-Kwong Lo. (Shortened version) Alice and Bob agree on a random permutation of the bits in the key They split the key into blocks of length k Compare the parity of each block. If they compute the same parity, the block is considered correct. If their parity is different, they look for the erroneous bit, using a binary search in the block. Alice and Bob discard the last bit of each block whose parity has been announced This is repeated with different permutations and block size, until Alice and Bob fail to find any disagreement in many subsequent comparisons
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Privacy amplification Eve might have partial knowledge of the key. Transform the key into a shorter but secure key Suppose there are n bits in the key and Eve has knowledge of m bits. Randomly chose a hash function whereh(x): {0,1}n {0,1} n-m-s Reduces Eve's knowledge of the key to 2 –s / ln2 bits
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Binary information A user can suggest a key by sending a stream of randomly polarized photons. This sequence can be converted to a binary key. If the key was intercepted it could be discarded and a new stream of randomly polarized photons sent.
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It solved thekey distribution problem. Unconditionally secure key distribution method proposed by: Charles Bennett and Gilles Brassard in 1984. . The method is called BB84. Once key is securely received it can be used to encrypt messages transmitted by conventional channels. The Main contribution of Quantum Cryptography.
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Quantum cryptography obtains its fundamental security from the fact that each qubit is carried by a single photon, and each photon will be altered as soon as it is read. This makes impossible to intercept message without being detected. Security of quantum key distribution
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Experimental implementations have existed since 1990. Current (2004) QC is performed over distances of 30-40 kilometers using optical fiber. In general we need two capabilities. Single photon gun. (2) Being able to measure single photons. State of the Quantum Cryptography technology.
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id Quantique, Geneva Switzerland Optical fiber based system Tens of kilometers distances MagiQ Technologies, NY City Optical fiber-glass Up to 100 kilometers distances NEC Tokyo 150 kilometers QinetiQ Farnborough, England Through the air 10 kilometers. Supplied system to BBN in Cambridge Mass. Commercial QC providers
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References [WIK2] Wikipedia -The free encyclopediahttp://www.wikipedia.org/wiki/Interpretation_of_quantum_mechanics [WIK3] Wikipedia -The free encyclopediahttp://www.wikipedia.org/wiki/Copenhagen_interpretation [GIT] Georgia Institute of Technology,The fundamental postulates of quantum mechanicshttp://www.physics.gatech.edu/academics/Classes/spring2002/6107/Resources/The fundamental postulates of quantum mechanics.pdf [HP] Hoi-Kwong Lo, Networked Systems Department,Hewlett Packard, Bristol, December 1997, Quantum Cryptology [SS99] Simon Singh, Code Book, p349-382,Anchor Books, 1999 [FoF] Forskning och Framsteg,No. 3, April 2003
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