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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Informed guessing of an eavesdropper's Renyi entropy

John M. Myers, Tai Tsun Wu

Published Aug 5, 2003• 0 citations
Publisher

Abstract

Users of a quantum cryptographic system face a problem of deciding on the ignorance of a maximally adroit eavesdropper concerning their key material. It is known that there can be no sure, positive, lower bound on any plausible measure of ignorance, and for this reason we characterize the problem as the making of an informed guess, meaning a guess that employs a rule that can be shown to work except in unlikely cases. As the measure of an eavesdropper's ignorance concerning n bits of sifted key material less some number k of bits found in error and discarded, we analyze Renyi entropy of arbitrary order R, for 1 ≤ R ≤ 2. We offer a rule for deciding on Renyi entropy based on a tighter bound on the relevant probability distributions than has been available. To this end, we employ a recently derived approximation to the cumulative binomial distribution which is uniformly accurate over a larger domain than previously available approximations. This results in a longer distilled key than that obtained from looser bounds, as well as generalizing the order R. Some numerical examples are presented.

DOI
10.1117/12.486994

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Users of a quantum cryptographic system face a problem of deciding on the ignorance of a maximally adroit eavesdropper concerning their key material. It is known that there can be no sure, positive, lower bound on any plausible measure of ignorance, and for this reason we characterize the problem as the making of an informed guess, meaning a guess that employs a rule that can be shown to work except in unlikely cases. As the measure of an eavesdropper's ignorance concerning n bits of sifted key material less some number k of bits found in error and discarded, we analyze Renyi entropy of arbitrary order R, for 1 ≤ R ≤ 2. We offer a rule for deciding on Renyi entropy based on a tighter bound on the relevant probability distributions than has been available. To this end, we employ a recently derived approximation to the cumulative binomial distribution which is uniformly accurate over a larger domain than previously available approximations. This results in a longer distilled key than that obtained from looser bounds, as well as generalizing the order R. Some numerical examples are presented.
The security of quantum key distribution against undetected eavesdropping depends on the key-sharing parties (Alice and Bob) making a probabilistic estimate of the ignorance of a maximally adept eavesdropper (Eve) concerning sifted, error-free bits from which Alice and Bob distill a key. For individual attacks on the BB84 protocol, we show how to generalize the defense function and the defense frontier of Slutsky et al. to take advantage of Cachin’s analysis of Renyi entropy of arbitrary order R, here called R-entropy. For a special case of an attack uniform over all bits, an optimum defense frontier is displayed. Evidence is discussed for the conjecture that this defense frontier in terms of R-entropy holds good not just for uniform attacks but for all individual attacks on BB84. We also show how the entropy estimate fits in to the full suite of key-distillation protocols in a QKD system, in particular how it relates to privacy amplification. After privacy amplification, Eve will have, with high probability, no information about the remaining bits. By choosing the optimal Rényi order R, we can distill secure bits in the presence of a significantly higher error rate.
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 of key sizes is derived based on conditional Rényi entropies. Our model and lower bounds include the Shannon’s perfect secrecy, and the minentropy based encryption presented by Dodis, and Alimomeni and Safavi-Naini. Finally, new optimal symmetric key cryptography and almost optimal secret sharing schemes are proposed which achieve our lower bounds.
Abstract Given a communication system using quantum key distribution (QKD), the receiver can be seen as one who tries to guess the sender’s information just as potential eavesdroppers do. The receiver-eavesdropper similarity thus implies a simple relation in terms of guessing probability and correctness of sifted keys, related with the distance-based, information-theoretic security. The tolerable regions of error rates determined by such a guessing-probability-based relation are shown to be close to those determined by security criteria. Thus, an alternative perspective on applying guessing probability in analyzing QKD issues is here provided. Examples of two specific protocols are illustrated. Our results contribute to evaluating an important element in communication study, and may provide useful reference for the security analysis of QKD protocols.
A novel definition of the conditional smooth Rényi entropy, which is different from that of Renner and Wolf, is introduced. It is shown that our definition of the conditional smooth Rényi entropy is appropriate for providing lower and upper bounds on the optimal guessing moment in a guessing problem where the guesser is allowed to stop guessing and declare an error. Further a general formula for the optimal guessing exponent is presented. In particular, a single-letterized formula for a mixture of i.i.d. sources is obtained. It is also shown that our definition is appropriate to characterize the optimal exponential moment of the codeword length in the problem of source coding with common side-information available at the encoder and decoder under a constraint on the probability of a decoding error.
Deriving formulations to estimate worst-case size bounds for conjunctive queries under various constraints has been at the core of theoretical database research. If the problem has no constraints or has a single functional dependency, tight worst-case size bounds are computable. If the problem has more than one functional dependency, computing tight bounds can be difficult in practice and may even require an infinite number of linear inequalities in its optimization formulation. While these challenges have been addressed with varying methods, no prior research has employed quantum information theory to address this problem. In this work, we establish a connection between earlier classical information theory-based works and quantum information theory. We propose replacing the classical Shannon entropy formulation with the quantum Rényi entropy of order α∈(0,1)α\in (0,1)α∈(0,1) whose entropy cone is characterized simply by non-negativity. The first key result is to express the bound in terms of optimizing over quantum states and Rényi entropy. Optimizing with respect to quantum states rather than classical distributions transfers the hardness into the problem of characterizing classical states, yielding a sound but generally not tight upper bound. We further quantify this hardness explicitly by proposing a dichotomy theorem: if the query satisfies a head absorption rule, then the Rényi program value is at most one and ∣Q(D)∣≤rmax(D)|Q(D)| \leq \mathrm{rmax}(D)∣Q(D)∣≤rmax(D). Otherwise, the program is unbounded.