Quantum Cryptography Prafulla Basavaraja CS 265 – Spring 2005.
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Quantum Cryptography Prafulla Basavaraja CS 265 – Spring 2005
Introduction Classical cryptography – relies on time complexity of certain mathematical operations given current computing methods Quantum computers – can solve these hard problems in polynomial time. –Example – Shor’s algorithm for finding prime factors of very large numbers Unbreakable code – One-Time pad –Practical difficulty is in key distribution Quantum Cryptography solves this problem by providing secure key distribution using quantum mechanics First suggested by C.Bennett and G.Brassard
Quantum Mechanics Deals with behavior of elementary particles (atoms & energy) in terms of probabilities Energy, momentum & angular momentums as well as charges come in discrete amounts called ‘quanta’ –Photons are discrete bundles of energy that make up light Properties that describe behavior of photon 1.Superposition principle Position or energy of the photon can simultaneously possess 2 or more values Photon Ray Gun experiment
Quantum Mechanics (contd.) 2.Entanglement property Applies to a pair of spatially separated photon where each is described with reference to other In case of entangled photons – measurement of spin of one gives the spin of other 3.Measurement problem Measuring the state of a photon changes it Polarization of Photons –Photon has electric and magnetic fields represented by vectors perpendicular to each other and direction of travel –Polarization describes the spin nature of a photon determined by electric vector of photon
Quantum State based Coding Scheme Polaroid – a filter to polarize & measure the polarization of photons Using rectilinear and diagonal polarization schemes gives 4 quantum bits or ‘qubits’ + 45 o 1 1 0 0 XX + We shall use the following notations: ‘+’ represents rectilinear scheme (horizontal and vertical polaroids) ‘-’ to represent 0 ‘|’ to represent 1 ‘x’ represents diagonal scheme (left and right inclined diagonal polariods) ‘/ ‘ to represent 0 ‘\’ to represent 1
Qubit transmission and binary digit selection - example Using the above qubit representations, a transmission for the binary 11010011 could look like this: Alice sends the 1 st 1 using the + scheme, the 2 nd one using the X scheme, 1 st 0 using the X scheme and so on. Alice : Scheme++++XXX+ Alice : Bits11010011 Alice : Qubit↕↕↔↕∕∕\↕ Bob : Scheme+XX++XX+ Bob : Qubit↕\\↕↔∕\↕ Bob : Bits11110011 Key Selection√√√√√
BB84 protocol – Quantum Key Distribution Step 1: Alice transmits random seq of 1s & 0s (qubits) to Bob over quantum channel –Alice uses random selection of rectilinear and diagonal schemes –Bob also uses random schemes to detect polarization of received photons – so interprets the 1s & 0s correctly only sometimes Step 2: Over a regular channel Alice tells Bob the polarization scheme she used for each qubit –Bob tells Alice when he used the same scheme & notes down bits determined with right scheme Step 3: Out of bits selected Alice & Bob pick a small subset and compare if they got the bits right. Eg:100 out of 500 bits –If the bits match discard bits used to compare & use remaining as the key (for encrypting actual data) –If the bits do not match it could be due to Eve whose detector had modified the polarization of a photon in transmission. So discard all bits and restart from step 1.
Key Selection and Detecting Eve’s presence Bit Number12345678 Alice : Bits11010011 Alice : Scheme++++XXX+ Alice : Qubit↕↕↔↕∕∕\↕ Eve : SchemeX+X++X+X Eve : Qubit/↕↕↕↔/↕/ Bob : Scheme+XX++XX+ Bob : Qubit↕\\↕↔∕/↔ Bob : Bits11110000 Selection√√√√√ Here the bits 1, 4, 6, 7, 8 are selected by Alice and Bob since both of them use the same detection scheme. But when they randomly check bits 1, 7 and 8 they find that the values are different. Through this they can detect the presence of the eavesdropper.
Practical problems with QC Beam Splitting attack –Hard to produce beam of single photons –Eve can use beam splitter when multiple photons are emitted –However it is not easy for Eve to determine when multiple photons are emitted; Splitting single photon will affect the state of the photon and give away Eve’s presence Man in the middle attack –Bob & Alice need proper authentication before talking to each other Distance limitation and media limitation –Fibre optics Optical pulse travels limited distance with out amplification – so have to be done hop by hop. Distance achieved - 87 kms. –Open space communication higher error rate 20 – 30 kms