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By Fan T.-H.

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We make use of information theory and refer to [66] for an introduction to the subject. 2 Information-Theoretic Security: Perfect Secrecy and Shannon’s Theorem Let us start with the classical scenario of a symmetric cryptosystem with message M , key K, and ciphertext C (see Fig. 1). The following security definition appears to be the strongest possible for such a cryptosystem. 1 [252]. , if I(M ; C) = 0 holds). Equivalent characterizations of this condition are that M and C are statistically independent or that the best strategy of an eavesdropper who wants to obtain (information about) the message from the ciphertext is to use only the a priori knowledge about M and to discard C.

6 Protocol Monotones and Upper Bounds The described protocol techniques lead to lower bounds on the quantity of interest, the secret-key rate S. One is, on the other hand, interested in upper bounds on S and, ultimately, determining S precisely; the latter, however, has been successfully done in trivial cases only in the two-way-communication setting. Characterization of the One-Way Key Rate In contrast to this, the one-way communication scenario has been completely solved [73]. 9. Let X, Y , and Z be random variables with joint distribution PXY Z .

Maurer et al. Because F is chosen from a two-universal family of functions with range U, we have EF [δF (x),F (x′ ) ] ≤ 1/|U|, for any x = x′ . Hence, EF PF (X) 2 2 Pr[E]2 Pr[E]2 ≤ max PEX (x) + . 7) concludes the proof. 3. Let X and Z be random variables and let F be chosen at random from a two-universal family of functions from X to U, where |U| = 2ℓ . Then, for any ε ≥ 0, PF (X)ZF − PU × PZ × PF 1 ≤ 2− (Hεmin (X|Z) − ℓ)/2 + 2ε, where PU is the uniform distribution on U. 2. Note that the distinguishing probability between a perfect key U and the function output F (X) is given by half the L1 -distance on the left-hand side of the corollary.

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A Bayesian analysis for the seismic data on Taiwan by Fan T.-H.

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