Distributions Attaining Secret Key at a Rate of the Conditional Mutual Information

  • Eric Chitambar
  • Benjamin Fortescue
  • Min-Hsiu Hsieh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9216)


In this paper we consider the problem of extracting secret key from an eavesdropped source \(p_{XYZ}\) at a rate given by the conditional mutual information. We investigate this question under three different scenarios: (i) Alice (X) and Bob (Y) are unable to communicate but share common randomness with the eavesdropper Eve (Z), (ii) Alice and Bob are allowed one-way public communication, and (iii) Alice and Bob are allowed two-way public communication. Distributions having a key rate of the conditional mutual information are precisely those in which a “helping” Eve offers Alice and Bob no greater advantage for obtaining secret key than a fully adversarial one. For each of the above scenarios, strong necessary conditions are derived on the structure of distributions attaining a secret key rate of I(X : Y|Z). In obtaining our results, we completely solve the problem of secret key distillation under scenario (i) and identify H(S|Z) to be the optimal key rate using shared randomness, where S is the Gács-Körner Common Information. We thus provide an operational interpretation of the conditional Gács-Körner Common Information. Additionally, we introduce simple example distributions in which the rate I(X : Y|Z) is achievable if and only if two-way communication is allowed.


Information-theoretic security Public key agreement Gács-Körner Common Information 


  1. 1.
    Ahlswede, R., Csiszár, I.: Common randomness in information theory and cryptography. i. secret sharing. IEEE Trans. Inf. Theory 39(4), 1121–1132 (1993)Google Scholar
  2. 2.
    Bennett, C., Brassard, G., Crepeau, C., Maurer, U.: Generalized privacy amplification. IEEE Trans. Inf. Theory 41(6), 1915–1923 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Christandl, M., Renner, R., Wolf, S.: A property of the intrinsic mutual information. In: Proceedings of the IEEE International Symposium on Information Theory 2003, pp. 258–258, June 2003Google Scholar
  4. 4.
    Csiszár, I., Narayan, P.: Common randomness and secret key generation with a helper. IEEE Trans. Inf. Theory 46(2), 344–366 (2000)CrossRefzbMATHGoogle Scholar
  5. 5.
    Csiszár, I., Körner, J.: Information Theory: Coding Theorems for Discrete Memoryless Systems. Cambridge University Press, Cambridge (2011)CrossRefGoogle Scholar
  6. 6.
    Gács, P., Körner, J.: Common information is far less than mutual information. Probl. Control Inf. Theory 2(2), 149 (1973)zbMATHGoogle Scholar
  7. 7.
    Gohari, A., Anantharam, V.: Information-theoretic key agreement of multiple terminals; part i. IEEE Trans. Inf. Theory 56(8), 3973–3996 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Maurer, U.: Secret key agreement by public discussion from common information. IEEE Trans. Inf. Theory 39(3), 733–742 (1993)CrossRefzbMATHGoogle Scholar
  9. 9.
    Maurer, U., Wolf, S.: Unconditionally secure key agreement and the intrinsic conditional information. IEEE Trans. Inf. Theory 45(2), 499–514 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ozols, M., Smith, G., Smolin, J.A.: Bound entangled states with a private key and their classical counterpart. Phys. Rev. Lett. 112, 110502 (2014)CrossRefGoogle Scholar
  11. 11.
    Renner, R., Wolf, S.: New bounds in secret-key agreement: the gap between formation and secrecy extraction. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 562–577. Springer, Heidelberg (2003) CrossRefGoogle Scholar

Copyright information

© International Association for Cryptologic Research 2015

Authors and Affiliations

  • Eric Chitambar
    • 1
  • Benjamin Fortescue
    • 1
  • Min-Hsiu Hsieh
    • 2
  1. 1.Department of Physics and AstronomySouthern Illinois UniversityCarbondaleUSA
  2. 2.Faculty of Engineering and Information Technology (FEIT), Centre for Quantum Computation & Intelligent Systems (QCIS)University of Technology Sydney (UTS)SydneyAustralia

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