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Classical and quantum security analysis via smoothing of Renyi entropy of order 2

Masahito Hayashi

Published Feb 1, 2012• 5 citations
OpenAlex

Abstract

It is known that the security evaluation can be done by smoothing of Renyi entropy of order 2 in the classical ans quantum settings when we apply universal2 hash functions. This fact can be extended to the case when we apply ǫ-almost dual universal2 hash functions. Demonstrating the smoothing of Renyi entropy of order 2, we derived security bounds for universal composability and mutual information criterion under the condition in the classical and quantum setting. Further, we apply this analysis to the secret key generation with error correction.

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It is known that the security evaluation can be done by smoothing of Renyi entropy of order 2 in the classical ans quantum settings when we apply universal2 hash functions. This fact can be extended to the case when we apply ǫ-almost dual universal2 hash functions. Demonstrating the smoothing of Renyi entropy of order 2, we derived security bounds for universal composability and mutual information criterion under the condition in the classical and quantum setting. Further, we apply this analysis to the secret key generation with error correction.
We introduce an improved one-shot characterisation of randomness extraction against quantum side information (privacy amplification), strengthening known one-shot bounds and providing a unified derivation of the tightest known asymptotic constraints. Our main tool is a new class of smooth conditional entropies defined by lifting classical smooth divergences through measurements. A key role is played by the measured smooth Rényi relative entropy of order 2, which we show to admit an equivalent variational form: it can be understood as allowing for smoothing over not only states, but also non-positive Hermitian operators. Building on this, we establish a tightened leftover hash lemma, significantly improving over all known smooth min-entropy bounds on extractable randomness and recovering the sharpest classical achievability results. We extend these methods to decoupling, the coherent analogue of privacy amplification, obtaining a corresponding improved one-shot bound. Relaxing our smooth entropy bounds leads to one-shot achievability results in terms of measured Rényi divergences, tightening the bounds of [Dupuis, arXiv:2105.05342] and recovering state-of-the-art asymptotic i.i.d. error exponents. We show an approximate optimality of our results by giving a matching one-shot converse bound up to additive logarithmic terms. This yields an optimal second-order asymptotic expansion of privacy amplification under trace distance, establishing a significantly tighter one-shot achievability result than previously shown in [Shen et al., arXiv:2202.11590] and proving its optimality for all hash functions.
It is known that the security evaluation can be done by smoothing of Rényi entropy of order 2 in the classical and quantum settings when we apply universal2hash functions. Using the smoothing of Rényi entropy of order 2, we derive security bounds for L1distinguishability and modified mutual information criterion under the classical and quantum setting, and have derived these exponential decreasing rates. These results are extended to the case when we apply ε-almost dual universal2hash functions. Furthermore, we apply this analysis to the secret key generation with error correction.
Motivated by the desirability of universal composability, we analyze in terms of L1 distinguishability the task of secret key generation from a joint random variable. Under this secrecy criterion, using the Rényi entropy of order 1+s for s ∈ [0,1], we derive a new upper bound of Eve's distinguishability under the application of the universal2 hash functions. It is also shown that this bound gives the tight exponential rate of decrease in the case of independent and identical distributions. The result is applied to the wiretap channel model and to secret key generation (distillation) by public discussion.
In this paper, we introduce the concept of dual universality of hash functions and present its applications to quantum cryptography. We begin by establishing the one-to-one correspondence between a linear function familyFand a code familyC, and thereby defining ε-almost dual universal2hash functions, as a generalization of the conventional universal2hash functions. Then, we show that this generalized (and thus broader) class of hash functions is in fact sufficient for the security of quantum cryptography. This result can be explained in two different formalisms. First, by noting its relation to the δ-biased family introduced by Dodis and Smith, we demonstrate that Renner's two-universal hashing lemma is generalized to our class of hash functions. Next, we prove that the proof technique by Shor and Preskill can be applied to quantum key distribution (QKD) systems that use our generalized class of hash functions for privacy amplification. While Shor-Preskill formalism requires an implementer of a QKD system to explicitly construct a linear code of the Calderbank-Shor-Steane (CSS) type, this result removes the existing difficulty of the construction of a linear code of CSS code by replacing it by the combination of an ordinary classical error correcting code and our proposed hash function. We also show that a similar result applies to the quantum wire-tap channel. Finally, we compare our results in the two formalisms and show that, in typical QKD scenarios, the Shor-Preskill-type argument gives better security bounds in terms of the trace distance and Holevo information than the method based on the δ-biased family.
The max-relative entropy together with its smoothed version is a basic tool in quantum information theory. In this paper, we derive the exact exponent for the asymptotic decay of the small modification of the quantum state in smoothing the max-relative entropy based on purified distance. We then apply this result to the problem of privacy amplification against quantum side information, and we obtain an upper bound for the exponent of the asymptotic decreasing of the insecurity, measured using either purified distance or relative entropy. Our upper bound complements the earlier lower bound established by Hayashi, and the two bounds match when the rate of randomness extraction is above a critical value. Thus, for the case of high rate, we have determined the exact security exponent. Following this, we give examples and show that in the low-rate case, neither the upper bound nor the lower bound is tight in general. This exhibits a picture similar to that of the error exponent in channel coding. Lastly, we investigate the asymptotics of equivocation and its exponent under the security measure using the sandwiched Rényi divergence of orders∈(1,2]s\in (1,2]s∈(1,2], which has not been addressed previously in the quantum setting.