Equivocations, Exponents, and Second-Order Coding Rates Under Various Rényi Information Measures

We evaluate the asymptotics of equivocations, their exponents as well as their second-order coding rates under various Rényi information measures. Specifically, we consider the effect of applying a hash function on a source and we quantify the level of non-uniformity and dependence of the compressed...

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Bibliographic Details
Published in:IEEE transactions on information theory 2017-02, Vol.63 (2), p.975-1005
Main Authors: Hayashi, Masahito, Tan, Vincent Y. F.
Format: Article
Language:English
Subjects:
Arimoto’s mutual information
Asymptotic methods
Asymptotic properties
Atmospheric measurements
Channel coding
Coding
conditional Rényi entropies
Correlation analysis
Cybersecurity
Dependence
Entropy
equivocation
error exponents
Exponents
Information theory
Information-theoretic security
Nonuniformity
Particle measurements
Random variables
Rényi divergence
second-order coding rates
secrecy exponents
Security
Sibson’s mutual information
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Summary:We evaluate the asymptotics of equivocations, their exponents as well as their second-order coding rates under various Rényi information measures. Specifically, we consider the effect of applying a hash function on a source and we quantify the level of non-uniformity and dependence of the compressed source from another correlated source when the number of copies of the sources is large. Unlike previous works that use Shannon information measures to quantify randomness, information, or uniformity, we define our security measures in terms of a more general class of information measures-the Rényi information measures and their Gallager-type counterparts. A special case of these Rényi information measure is the class of Shannon information measures. We prove tight asymptotic results for the security measures and their exponential rates of decay. We also prove bounds on the second-order asymptotics and show that these bounds match when the magnitudes of the second-order coding rates are large. We do so by establishing new classes non-asymptotic bounds on the equivocation and evaluating these bounds using various probabilistic limit theorems asymptotically.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2016.2636154