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arXiv:1607.01797 · SIAM Journal on Computing

Simple and Tight Device-Independent Security Proofs

Rotem Arnon-Friedman, Renato Renner, Thomas Vidick

Published Jan 1, 2019• 155 citations
arXivHTMLPDF

Abstract

Device-independent security is the gold standard for quantum cryptography: not only is security based entirely on the laws of quantum mechanics, but it holds irrespective of any a priori assumptions on the quantum devices used in a protocol, making it particularly applicable in a quantum-wary environment. While the existence of device-independent protocols for tasks such as randomness expansion and quantum key distribution has recently been established, the underlying proofs of security remain very challenging, yield rather poor key rates, and demand very high quality quantum devices, thus making them all but impossible to implement in practice. We introduce a technique for the analysis of device-independent cryptographic protocols. We provide a flexible protocol and give a security proof that provides quantitative bounds that are asymptotically tight, even in the presence of general quantum adversaries. At a high level our approach amounts to establishing a reduction to the scenario in which the untrusted device operates in an identical and independent way in each round of the protocol. This is achieved by leveraging the sequential nature of the protocol and makes use of a newly developed tool, the “entropy accumulation theorem” of Dupuis, Fawzi, and Renner [ Entropy Accumulation, preprint, 2016]. As concrete applications we give simple and modular security proofs for device-independent quantum key distribution and randomness expansion protocols based on the CHSH inequality. For both tasks, we establish essentially optimal asymptotic key rates and noise tolerance. In view of recent experimental progress, which has culminated in loophole-free Bell tests, it is likely that these protocols can be practically implemented in the near future.

DOI
10.1137/18m1174726

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Device-independent security is the gold standard for quantum cryptography: not only is security based entirely on the laws of quantum mechanics, but it holds irrespective of any a priori assumptions on the quantum devices used in a protocol, making it particularly applicable in a quantum-wary environment. While the existence of device-independent protocols for tasks such as randomness expansion and quantum key distribution has recently been established, the underlying proofs of security remain very challenging, yield rather poor key rates, and demand very high quality quantum devices, thus making them all but impossible to implement in practice. We introduce a technique for the analysis of device-independent cryptographic protocols. We provide a flexible protocol and give a security proof that provides quantitative bounds that are asymptotically tight, even in the presence of general quantum adversaries. At a high level our approach amounts to establishing a reduction to the scenario in which the untrusted device operates in an identical and independent way in each round of the protocol. This is achieved by leveraging the sequential nature of the protocol and makes use of a newly developed tool, the “entropy accumulation theorem” of Dupuis, Fawzi, and Renner [ Entropy Accumulation, preprint, 2016]. As concrete applications we give simple and modular security proofs for device-independent quantum key distribution and randomness expansion protocols based on the CHSH inequality. For both tasks, we establish essentially optimal asymptotic key rates and noise tolerance. In view of recent experimental progress, which has culminated in loophole-free Bell tests, it is likely that these protocols can be practically implemented in the near future.
Device-independent cryptography goes beyond conventional quantum cryptography by providing security that holds independently of the quality of the underlying physical devices. Device-independent protocols are based on the quantum phenomena of non-locality and the violation of Bell inequalities. This high level of security could so far only be established under conditions which are not achievable experimentally. Here we present a property of entropy, termed "entropy accumulation", which asserts that the total amount of entropy of a large system is the sum of its parts. We use this property to prove the security of cryptographic protocols, including device-independent quantum key distribution, while achieving essentially optimal parameters. Recent experimental progress, which enabled loophole-free Bell tests, suggests that the achieved parameters are technologically accessible. Our work hence provides the theoretical groundwork for experimental demonstrations of device-independent cryptography.
In device-independent (DI) quantum protocols, security statements are agnostic to the internal workings of the quantum devices-they rely solely on classical interactions with the devices and specific assumptions. Traditionally, such protocols are set in a non-local scenario, where two non-communicating devices exhibit Bell inequality violations. Recently, a new class of DI protocols has emerged that requires only a single device. In this setting, the assumption of no communication is replaced by a computational one: the device cannot solve certain post-quantum cryptographic problems. Protocols developed in this single-device computational setting-such as for randomness certification-have relied on ad hoc techniques, making their guarantees difficult to compare and generalize. In this work, we introduce a modular proof framework inspired by techniques from the non-local DI literature. Our approach combines tools from quantum information theory, including entropic uncertainty relations and the entropy accumulation theorem, to yield both conceptual clarity and quantitative security guarantees. This framework provides a foundation for systematically analyzing DI protocols in the single-device setting under computational assumptions. It enables the design and security proof of future protocols for DI randomness generation, expansion, amplification, and key distribution, grounded in post-quantum cryptographic hardness.
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Quantum key distribution (QKD) can be proved to be secure by laws of quantum mechanics. In this thesis, we review security proof methods in Renner's framework and discuss numerical methods to calculate asymptotic and finite key rates. These methods are highly versatile and applicable to general device-dependent QKD protocols. We also discuss analytical tools that extend the applicability of these numerical methods. We then present the asymptotic security proof against collective attacks for a variant of the twin-field QKD protocol, which can overcome the repeaterless secret-key capacity bound. Our variant reduces the sifting cost and uses non-phase-randomized coherent states as both signals and test states. We confirm the loss scaling of this protocol. Another important family of protocols that we investigate here are discrete-modulated continuous-variable QKD protocols. They are interesting due to their experimental simplicity and their great potential for massive deployment in the quantum-secured networks. Our security proof method can provide tight asymptotic key rates. We demonstrate that the postselection of data in combination with reverse reconciliation can improve the key rates. We analyze both untrusted and trusted detector noise scenarios. Our results in the trusted detector noise scenario show that we can thus cut out most of the effect of detector noise and obtain asymptotic key rates similar to those had we access to ideal detectors. Finally, we present several simple examples to illustrate our newly developed method for the numerical finite-key analysis against the most general attacks via the entropy accumulation theorem.
Security proofs in quantum cryptography rely on conditional entropies. In a many-round protocol, their estimation is a challenging task; one must account for the most general attacks by an eavesdropper, including those that are not independently and identically distributed (i.i.d.) across all rounds. Chain rules address this problem by relating the conditional entropy of a structured, but non-i.i.d. process to a sum of entropy contributions from each round. They are a key ingredient in entropy accumulation theorems (EATs), which provide a versatile security proof framework for many protocols in quantum cryptography. Recently, chain rules in the setting of trusted devices have lead to tight i.i.d. reductions at a finite number of rounds, and whether analogous results can be recovered in the device-independent (DI) setting has not been addressed. Surprisingly, we show that a natural tightening of the chain rule of Dupuis et al. [Commun. Math. Phys. 379, 867-913, (2020)] that would answer this question affirmatively cannot hold, highlighting a limitation of the current DI security proof approach. Nonetheless, we show that an intermediate improvement is possible by proving a new chain rule in this setting. Following the framework of Arqand et al. [Phys. Rev. X 15, 041013 (2025)], we use our chain rule to provide a slightly tighter version of the Rényi EAT in certain contexts. In addition, we provide a self-contained framework that unifies existing chain rules and compares their applications, framing our results in a broader context.
Device-independent (DI) cryptography represents the highest level of security, enabling cryptographic primitives to be executed safely on uncharacterized devices. Moreover, with successful proof-of-concept demonstrations in randomness expansion, randomness amplification, and quantum key distribution, the field is steadily advancing toward commercial viability. Critical to this continued progression is the development of tighter finite-size security proofs. In this work, we provide a simple method to obtain tighter finite-size security proofs for protocols based on the CHSH game, which is the nonlocality test used in all of the proof-of-concept experiments. We achieve this by analytically solving key-rate optimization problems based on Rényi entropies, providing a simple method to obtain tighter finite-size key rates.