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 Alice and the other to Bob

5. Bell’s Inequality Contd…  We will need to perform measurements of 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.  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:

9. Back to EPR QKD  A&B measure the remaining EPR pairs in 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 a 1, a 2 in A, the probability that g(a 1 )=g(a 2 ) is at most 1/|B|

14. Privacy Amplification Contd…  Now, both A&B compute S = G(W)  Collision Entropy of a random variable X is 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