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Rate-distortion Theory for Secrecy Systems

Rate-distortion Theory for Secrecy Systems. Paul Cuff Electrical Engineering Princeton University. Information Theory. Channel Coding. Source Coding. Secrecy. Secrecy. Channel. Source. Source Coding. Describe an information signal ( source ) with a message. Information.

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Rate-distortion Theory for Secrecy Systems

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  1. Rate-distortion Theory for Secrecy Systems Paul Cuff Electrical Engineering Princeton University

  2. Information Theory Channel Coding Source Coding Secrecy Secrecy Channel Source

  3. Source Coding • Describe an information signal (source) with a message. Information Reconstruction Message Encoder Decoder

  4. Entropy • If Xn is i.i.d. according to pX • R > H(X) is necessary and sufficient for lossless reconstruction Space of Xn sequences Enumerate the typical set

  5. Many Methods • For lossless source coding, the encoding method is not so important • It should simply use the full entropy of the bits

  6. Single Letter Encoding (method 1) • Encode each Xi separately • Under the constraints of decodability, Huffman codes are optimal • Expected length is within one bit of entropy • Encode tuples of symbols to get closer to the entropy limit

  7. Random Binning(method 2) • Assign to each Xn sequence a random bit sequence (hash function) 0100110101011 Space of Xn sequences 0110100010010 1101011000100

  8. Linear Transformation(method 3) Source Message Random Matrix J Xn

  9. Summary • For lossless source coding, structure of communication doesn’t matter much Information Gathered H(Xn) Message Bits Received

  10. Lossy Source Coding • What if the decoder must reconstruct with less than complete information? • Error probability will be close to one • Distortion as a performance metric

  11. Poor Performance • Random Binning and Random Linear Transformations are useless! Distortion E d(X,y) Time Sharing Massey Conjecture: Optimal for linear codes Message Bits Received

  12. Puzzle • Describe an n-bit random sequence • Allow 1 bit of distortion • Send only 1 bit

  13. Rate Distortion Theorem • [Shannon] • Choose p(y|x):

  14. Structure of Useful Partial Information • Coordination (Given source PX construct Yn ~ PY|X ) • Empirical • Strong

  15. Empirical Coordination Codes • Codebook • Random subset of Yn sequences • Encoder • Find the codeword that has the right joint first-order statistics with the source

  16. Strong Coordination PY|X • Black box acts like a memoryless channel • X and Y are an i.i.d. multisource Communication Resources Output Source

  17. Strong Coordination Synthetic Channel PY|X • Related to: • Reverse Shannon Theorem [Bennett et. al.] • Quantum Measurements [Winter] • Communication Complexity [Harsha et. al.] • Strong Coordination [C.-Permuter-Cover] • Generating Correlated R.V. [Anantharam, Gohari, et. al.] Common Randomness Message Output Source Node A Node B

  18. Structure of Strong Coord. K

  19. Information Theoretic Security

  20. Wiretap Channel[Wyner 75]

  21. Wiretap Channel[Wyner 75]

  22. Wiretap Channel[Wyner 75]

  23. Confidential Messages[Csiszar, Korner 78]

  24. Confidential Messages[Csiszar, Korner 78]

  25. Confidential Messages[Csiszar, Korner 78]

  26. Merhav 2008

  27. Villard-Piantanida 2010

  28. Other Examples of“rate-equivocation” theory • Gunduz-Erkip-Poor 2008 • Lia-H. El-Gamal 2008 • Tandon-Ulukus-Ramchandran 2009 • …

  29. Rate-distortion theory (secrecy)

  30. Achievable Rates and Payoff Given [Schieler, Cuff 2012 (ISIT)]

  31. How to Force High Distortion • Randomly assign bins • Size of each bin is • Adversary only knows bin • Adversary has no knowledge of only knowledge of

  32. Causal Disclosure

  33. Causal Disclosure (case 1)

  34. Causal Disclosure (case 2)

  35. Example • Source distribution is Bernoulli(1/2). • Payoff: One point if Y=X but Z≠X.

  36. Rate-payoff Regions

  37. General Disclosure Causal or non-causal

  38. Strong Coord. for Secrecy Channel Synthesis Information Action Node A Node B Attack Adversary Not optimal use of resources!

  39. Strong Coord. for Secrecy Channel Synthesis Information Action Node A Node B Un Attack Adversary Reveal auxiliary Un “in the clear”

  40. Payoff-Rate Function • Maximum achievable average payoff • Markov relationship: Theorem:

  41. Structure of Secrecy Code K

  42. Equivocation next Intermission

  43. Log-loss Distortion Reconstruction space of Z is the set of distributions.

  44. Best Reconstruction Yields Entropy

  45. Log-loss (disclose X causally)

  46. Log-loss (disclose Y causally)

  47. Log-loss (disclose X and Y)

  48. Result 1 from Secrecy R-D Theory

  49. Result 2 from Secrecy R-D Theory

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