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Information Theoretic Security for Encryption Based on Conditional Rényi Entropies

Part of the Lecture Notes in Computer Science book series (LNSC,volume 8317)

Abstract

In this paper, information theoretic cryptography is discussed based on conditional Rényi entropies. Our discussion focuses not only on cryptography but also on the definitions of conditional Rényi entropies and the related information theoretic inequalities. First, we revisit conditional Rényi entropies, and clarify what kind of properties are required and actually satisfied. Then, we propose security criteria based on Rényi entropies, which suggests us deep relations between (conditional) Rényi entropies and error probabilities by using several guessing strategies. Based on these results, unified proof of impossibility, namely, the lower bounds on key sizes are derived based on conditional Rényi entropies. Our model and lower bounds include the Shannon’s perfect secrecy, and the min-entropy based encryption presented by Dodis, and Alimomeni and Safavi-Naini at ICITS2012. Finally, a new optimal symmetric key encryption protocol achieving the lower bounds is proposed.

Keywords

  • Information theoretic cryptography
  • (Conditional) Rényi entropy
  • Error probability in guessing
  • Impossibility
  • Symmetric-key encryption

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Notes

  1. 1.

    Throughout of the paper, the base of logarithm is \(e\). Note that the base of logarithm is not essential since the same arguments hold for arbitrary base of logarithm. We also define \(0^0:=0\) for \(\alpha =0\).

  2. 2.

    This form of the chain rule is inductively obtained by using the postulate (d) in [8, p. 547].

  3. 3.

    In the case of \(\alpha = 1\), conditional Rényi entropies coincide with conditional Shannon entropy, and hence, chain rule is of course satisfied. In addition, it is obvious that \(R_\alpha ^\mathsf{JA}(X|Y)\) also satisfies the chain rule since it is defined to satisfy the chain rule.

  4. 4.

    We can show that CRE is satisfied by \(R_\alpha ^\mathsf{RW}(X|Y)\) in the case of \(\alpha > 1\).

  5. 5.

    Strictly speaking, our bounds are slightly more general than Shannon’s bounds and Alimomeni and Safavi-Naini’s one, since we have removed the assumption that \(\pi _{enc}\) and \(\pi _{dec}\) are deterministic

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Acknowledgments

The authors would like to thank the anonymous referees for their helpful comments. Mitsugu Iwamoto is supported by JSPS KAKENHI Grant No. 23760330. Junji Shikata is supported by JSPS KAKENHI Grant No. 23500012.

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Iwamoto, M., Shikata, J. (2014). Information Theoretic Security for Encryption Based on Conditional Rényi Entropies. In: Padró, C. (eds) Information Theoretic Security. ICITS 2013. Lecture Notes in Computer Science(), vol 8317. Springer, Cham. https://doi.org/10.1007/978-3-319-04268-8_7

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  • DOI: https://doi.org/10.1007/978-3-319-04268-8_7

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