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EECS 598 Fall ’01

EECS 598 Fall ’01. Quantum Cryptography Presentation By George Mathew. What’s been done so far (recap):. Introduction to Cryptosystems Quantum Properties just provides a new method for private key distribution Some QKD Protocols. Overview:. The EPR protocol for Quantum Key Distribution

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EECS 598 Fall ’01

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  1. EECS 598 Fall ’01 Quantum Cryptography Presentation By George Mathew

  2. What’s been done so far (recap): • Introduction to Cryptosystems • Quantum Properties just provides a new method for private key distribution • Some QKD Protocols

  3. Overview: • The EPR protocol for Quantum Key Distribution • Information Reconciliation • Privacy Amplification • Summary

  4. Bells Inequality: • Suppose we have 2 qubits in the state • One qubit is passed to Aliceand the other to Bob

  5. Bell’s Inequality Contd… • We will need to perform measurementsof the following observables:

  6. Bell’s Inequality Contd… • The average values for these observables: • Thus,

  7. Bell’s Inequality Contd… • But if they were classical bits: • this is a test for the fidelity of an EPR pair

  8. EPR Protocol for QKD: • 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

  9. 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 bits

  10. Privacy 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?

  11. 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 W

  12. Privacy 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.

  13. 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|

  14. Privacy Amplification Contd… • Now, both A&B compute S = G(W) • Collision Entropy of a random variable Xis defined as:

  15. Privacy Amplification Contd… • It can be shown that

  16. 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.

  17. 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 function

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