EECS 598 Fall ’01What’s been done so far (recap):Overview:Bells Inequality:Bell’s Inequality Contd…Slide 6Slide 7EPR Protocol for QKD:Back to EPR QKDPrivacy Amplification and Information ReconciliationInformation Reconciliation:Privacy AmplificationPrivacy Amplification Contd…Slide 14Slide 15Slide 16SummaryEECS 598 Fall ’01Quantum Cryptography PresentationQuantum Cryptography PresentationBy George MathewBy George MathewWhat’s been done so far (recap):Introduction to CryptosystemsQuantum Properties just provides a new method for private key distributionSome QKD ProtocolsOverview:The EPR protocol for Quantum Key DistributionInformation ReconciliationPrivacy AmplificationSummaryBells Inequality:Suppose we have 2 qubits in the stateOne qubit is passed to Aliceand the other to Bob21001 Bell’s Inequality Contd…We will need to perform measurementsof the following observables:222XZS222XZT1ZQ 1XR Bell’s Inequality Contd…The average values for these observables:Thus,21;21;21;21 QTRTRSQS22 QTRTRSQSBell’s Inequality Contd…But if they were classical bits:this is a test for the fidelity of an EPR pair       2 QTERTERSEQSEUses the properties of entanglement. Alice and Bob share a set of n EPR pairsThey select a random subset of the EPR pairs–Use communication over a public channel–Test for violation of Bell’s InequalityIf they don’t violate it–this places a lower bound on the fidelity of the remaining pairs EPR Protocol for QKD:21100 Back to EPR QKDA&B measure the remaining EPR pairsin jointly determined random basesThis gives them correlated classical bits, from which they can get secret key bitsPrivacy Amplification and Information ReconciliationA & B have done a QKD and now share correlated classical bit strings X and Y.X and Y are imperfect keys because of Eve and noiseHow do we “distill” a key good enough for a secure transaction?Information Reconciliation:Information reconciliation =error correction between X and Y over a public channelThus A &B obtain a shared bit-string WEve obtains Z, which is partially correlated with WPrivacy AmplificationPrivacy Amplification is used to get a smaller set of bits, S, from W, whose correlation with Z is below a certain threshold. How does it work??I tried… but I’m not very sure yet.Privacy Amplification Contd…Both Alice and Bob choose a random Universal Hash Function G.Definition: A universal hash function g maps an n-bit string A to an m-bit string B such that, given a1, a2 in A, the probability that g(a1)=g(a2) is at most 1/|B|Privacy Amplification Contd…Now, both A&B compute S = G(W)Collision Entropy of a random variable Xis defined as:xcxpXH2)(log)(Privacy Amplification Contd…It can be shown that2ln2),()( XHmcmzZGSHPrivacy Amplification Contd…m can be chosen small enough so that the entropy is almost equal to m. This maximizes Eve’s uncertainty about S.SummaryEPR Protocol: Uses Bell’s inequality to test for fidelityInformation Reconciliation: Error Correction between Alice’s and Bob’s bit stringsPrivacy Amplification: Reduce Eve’s information about key bits by using a universal hashing