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arXiv:1205.0842 · Physical Review A

Min-entropy uncertainty relation for finite-size cryptography

Nelly H. Y. Ng, Mario Berta, Stephanie Wehner

Published Oct 15, 2012• 58 citations
arXivHTMLPDF

Abstract

Apart from their foundational significance, entropic uncertainty relations play a central role in proving the security of quantum cryptographic protocols. Of particular interest are therefore relations in terms of the smooth min-entropy for Bennett-Brassard 1984 (BB84) and six-state encodings. The smooth min-entropy ${H}_{\mathrm{min}}^{\ensuremath{\epsilon}}(X/B)$ quantifies the negative logarithm of the probability for an attacker $B$ to guess $X$, except with a small failure probability $\ensuremath{\epsilon}$. Previously, strong uncertainty relations were obtained which are valid in the limit of large block lengths. Here, we prove an alternative uncertainty relation in terms of the smooth min-entropy that is only marginally less strong but has the crucial property that it can be applied to rather small block lengths. This paves the way for a practical implementation of many cryptographic protocols. As part of our proof we show tight uncertainty relations for a family of R\'enyi entropies that may be of independent interest.

DOI
10.1103/physreva.86.042315

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Apart from their foundational significance, entropic uncertainty relations play a central role in proving the security of quantum cryptographic protocols. Of particular interest are therefore relations in terms of the smooth min-entropy for Bennett-Brassard 1984 (BB84) and six-state encodings. The smooth min-entropy Hmin\ensuremathϵ(X/B){H}_{\mathrm{min}}^{\ensuremath{\epsilon}}(X/B)Hmin\ensuremathϵ​(X/B) quantifies the negative logarithm of the probability for an attacker BBB to guess XXX, except with a small failure probability \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ. Previously, strong uncertainty relations were obtained which are valid in the limit of large block lengths. Here, we prove an alternative uncertainty relation in terms of the smooth min-entropy that is only marginally less strong but has the crucial property that it can be applied to rather small block lengths. This paves the way for a practical implementation of many cryptographic protocols. As part of our proof we show tight uncertainty relations for a family of R'enyi entropies that may be of independent interest.
We derive a new entropic quantum uncertainty relation involving min-entropy. The relation is tight and can be applied in various quantum-cryptographic settings. Protocols for quantum 1-out-of-2 Oblivious Transfer and quantum Bit Commitment are presented and the uncertainty relation is used to prove the security of these protocols in the bounded quantum-storage model according to new strong security definitions. As another application, we consider the realistic setting of Quantum Key Distribution (QKD) against quantum-memory-bounded eavesdroppers. The uncertainty relation allows to prove the security of QKD protocols in this setting while tolerating considerably higher error rates compared to the standard model with unbounded adversaries. For instance, for the six-state protocol with one-way communication, a bit-flip error rate of up to 17% can be tolerated (compared to 13% in the standard model). Our uncertainty relation also yields a lower bound on the min-entropy key uncertainty against known-plaintext attacks when quantum ciphers are composed. Previously, the key uncertainty of these ciphers was only known with respect to Shannon entropy.
Any key-generation session contains a finite number of quantum-state messages, and it is there-fore important to understand the fundamental restrictions imposed on the minimal length of a string required to obtain a secret key with a specified length. The entropy uncertainty relations for smooth min and max entropies considerably simplify and shorten the proof of security. A proof of security of quantum key distribution with phase-temporal encryption is presented. This protocol provides the maximum critical error compared to other protocols up to which secure key distribution is guaranteed. In addition, unlike other basic protocols (of the BB84 type), which are vulnerable with respect to an attack by “blinding” of avalanche photodetectors, this protocol is stable with respect to such an attack and guarantees key security.
This thesis consolidates, improves and extends the smooth entropy framework for non-asymptotic information theory and cryptography. We investigate the conditional min- and max-entropy for quantum states, generalizations of classical Rényi entropies. We introduce the purified distance, a novel metric for unnormalized quantum states and use it to define smooth entropies as optimizations of the min- and max-entropies over a ball of close states. We explore various properties of these entropies, including data-processing inequalities, chain rules and their classical limits. The most important property is an entropic formulation of the asymptotic equipartition property, which implies that the smooth entropies converge to the von Neumann entropy in the limit of many independent copies. The smooth entropies also satisfy duality and entropic uncertainty relations that provide limits on the power of two different observers to predict the outcome of a measurement on a quantum system. Finally, we discuss three example applications of the smooth entropy framework. We show a strong converse statement for source coding with quantum side information, characterize randomness extraction against quantum side information and prove information theoretic security of quantum key distribution using an intuitive argument based on the entropic uncertainty relation.
We compare three proof techniques for composable finite-size security of quantum key distribution under collective attacks, with emphasis on how the resulting secret-key rates behave at practically relevant block lengths. As a benchmark, we consider the BB84 protocol and evaluate finite-size key-rate estimates obtained from entropic uncertainty relations (EUR), from the asymptotic equipartition property (AEP), and from a direct finite-block analysis based on the conditional min-entropy, which we refer to as the finite-size min-entropy (FME) approach. For BB84 we show that the EUR-based bound provides the most favorable performance across the considered parameter range, while the AEP bound is asymptotically tight but can become overly pessimistic at moderate and small block sizes, where it may fail to certify a positive key. The FME approach remains effective in this small-block regime, yielding nonzero rates in situations where the AEP estimate vanishes, although it is not asymptotically optimal for BB84. These results motivate the use of FME-type analyses for continuous-variable protocols in settings where tight EUR-based bounds are unavailable, notably for coherent-state schemes where current finite-size analyses typically rely on AEP-style corrections.
Entropic uncertainty relations are underpinning to compute the quantitative security bound in quantum cryptographic applications, such as quantum random number generation (QRNG) and quantum key distribution (QKD). All security proofs derive a relation between the information accessible to the legitimate group and the maximum knowledge that an adversary may have gained, Eve, which exploits entropic uncertainty relations to lower bound Eve's uncertainty about the raw key generated by one party, Alice. The standard entropic uncertainty relations is to utilize the smooth min- and max-entropies to show these cryptographic applications' security by computing the overlap of two incompatible measurements or positive-operator valued measures (POVMs). This paper draws one case of the POVM-versioned standard entropic uncertainty relation yielding the trivial bound since the maximum overlap in POVMs always produces the trivial value, "one." So, it fails to tie the smooth min-entropy to show the security of the quantum cryptographic application.
Despite enormous theoretical and experimental progress in quantum cryptography, the security of most current implementations of quantum key distribution is still not rigorously established. One significant problem is that the security of the final key strongly depends on the number, M, of signals exchanged between the legitimate parties. Yet, existing security proofs are often only valid asymptotically, for unrealistically large values of M. Another challenge is that most security proofs are very sensitive to small differences between the physical devices used by the protocol and the theoretical model used to describe them. Here we show that these gaps between theory and experiment can be simultaneously overcome by using a recently developed proof technique based on the uncertainty relation for smooth entropies. Although they offer significant promise, practical implementations of quantum key distribution are often not as rigorous as theory predicts. This study demonstrates how two instances of such discrepancies can be resolved by taking advantage of an enotropic formulation of the uncertainty principle.