Quantum key distribution (QKD) enables secure information exchange with untrusted quantum links between remote communication parties. A central task in the studies of QKD protocols is security analysis, which aims at deriving the length of the final key such that it is secure regardless of the eavesdropper's computational power. In literature, the security analysis has been done both analytically and numerically. Compared to analytical methods which tend to require techniques specific to each QKD protocol, numerical ones are more general since they can be directly applied to many QKD protocols with little adaptation. However, current numerical methods are carried out based on some assumptions such as working in the asymptotic limit and collective attacks from eavesdroppers. In this work, we remove these assumptions and develop a numerical finite-size security analysis against general attacks for a large class of QKD protocols. We also give an example of applying the method to the recent phase-matching QKD protocol with a simplified protocol design. Our result shows that the finite-size key rate can surpass the linear key-rate bound with a practical QKD system operated for only several hours.
Quantum key distribution (QKD) enables secure information exchange with untrusted quantum links between remote communication parties. A central task in the studies of QKD protocols is security analysis, which aims at deriving the length of the final key such that it is secure regardless of the eavesdropper's computational power. In literature, the security analysis has been done both analytically and numerically. Compared to analytical methods which tend to require techniques specific to each QKD protocol, numerical ones are more general since they can be directly applied to many QKD protocols with little adaptation. However, current numerical methods are carried out based on some assumptions such as working in the asymptotic limit and collective attacks from eavesdroppers. In this work, we remove these assumptions and develop a numerical finite-size security analysis against general attacks for a large class of QKD protocols. We also give an example of applying the method to the recent phase-matching QKD protocol with a simplified protocol design. Our result shows that the finite-size key rate can surpass the linear key-rate bound with a practical QKD system operated for only several hours.
Abstract Quantum key distribution (QKD) allows for secure communications safe against attacks by quantum computers. QKD protocols are performed by sending a sizeable, but finite, number of quantum signals between the distant parties involved. Many QKD experiments, however, predict their achievable key rates using asymptotic formulas, which assume the transmission of an infinite number of signals, partly because QKD proofs with finite transmissions (and finite-key lengths) can be difficult. Here we develop a robust numerical approach for calculating the key rates for QKD protocols in the finite-key regime in terms of two semi-definite programs (SDPs). The first uses the relation between conditional smooth min-entropy and quantum relative entropy through the quantum asymptotic equipartition property, and the second uses the relation between the smooth min-entropy and quantum fidelity. The numerical programs are formulated under the assumption of collective attacks from the eavesdropper and can be promoted to withstand coherent attacks using the postselection technique. We then solve these SDPs using convex optimization solvers and obtain numerical calculations of finite-key rates for several protocols difficult to analyze analytically, such as BB84 with unequal detector efficiencies, B92, and twin-field QKD. Our numerical approach democratizes the composable security proofs for QKD protocols where the derived keys can be used as an input to another cryptosystem.
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.
The goal of quantum key distribution (QKD) is to establish a secure key between two parties connected by an insecure quantum channel. To use a QKD protocol in practice, one has to prove that a finite size key is secure against general attacks: no matter the adversary's attack, they cannot gain useful information about the key. A much simpler task is to prove security against collective attacks, where the adversary is assumed to behave identically and independently in each round. In this work, we provide a formal framework for general QKD protocols and show that for any protocol that can be expressed in this framework, security against general attacks reduces to security against collective attacks, which in turn reduces to a numerical computation. Our proof relies on a recently developed information-theoretic tool called generalised entropy accumulation and can handle generic prepare-and-measure protocols directly without switching to an entanglement-based version.
This thesis is concerned with rigorous security analyses of practical Quantum Key Distribution (QKD) protocols, using a variety of modern proof techniques. The main results are as follows. First, we establish a security proof for variable-length QKD protocols against IID collective attacks, and extend this result to coherent attacks using the postselection technique. In doing so, we resolve a long-standing flaw in the application of the postselection technique to QKD, thereby placing it on a rigorous mathematical footing. Second, we develop a method to bound phase error rates in entropic uncertainty relation-based and phase error rate-based proofs, using only the observed statistics of the protocol, even when detectors are imperfect and only approximately characterized. This removes a key assumption of identical detector behaviour and enables these techniques to be applied in realistic settings. Third, we present a very general security analysis based on the marginal-constrained entropy accumulation theorem. The resulting framework can be readily adapted to practical imperfections and side channels, and is suitable for certification efforts. Finally, we show that the security of QKD protocols under realistic authentication assumptions can be reduced to the standard idealized setting, where authentication is assumed to behave honestly, with only minor protocol modifications. A distinctive feature of this thesis is its unified presentation of several major QKD security proof frameworks using consistent protocol descriptions and notation. Consequently, this thesis is intended not only as a collection of new technical results, but also as a useful reference for understanding rigorous security analysis in quantum key distribution.
The goal of quantum key distribution (QKD) is to establish a secure key between two parties connected by an insecure quantum channel. To use a QKD protocol in practice, one has to prove that a finite size key is secure against general attacks: no matter the adversary's attack, they cannot gain useful information about the key. A much simpler task is to prove security against collective attacks, where the adversary is assumed to behave identically and independently in each round. In this work, we provide a formal framework for general QKD protocols and show that for any protocol that can be expressed in this framework, security against general attacks reduces to security against collective attacks, which in turn reduces to a numerical computation. Our proof relies on a recently developed information-theoretic tool called generalised entropy accumulation and can handle generic prepare-and-measure protocols directly without switching to an entanglement-based version.
Quantum Key Distribution, or short QKD, aims to establish a secure key withoutmaking any additional assumptions about the abilities or computational power of an adversary who is only limited by the laws of nature. The process of finding a mathematical expression for, or at least a lower bound on, the secure keyrate of some QKD protocol, given relevant system parameters, is called securityproof. In this thesis, we use a recent numerical security proof technique to examine different post selection strategies for continuous-variable quantum key distribution(CV-QKD) protocols with quadrature phase-shift keying modulation and four or eight signal states. CV-QKD protocols use coherent states to encode information and measure the field-quadrature components by homodyne or heterodyne detection. The basic idea of the used numerical security proof technique is to solve thekey rate finding problem in a two-step process. In the first step, the problem is solved approximately, using a numerical algorithm, which yields an upper bound on the secure key rate. This is followed by step two, where the obtained upperbound is converted into a lower bound, using a sequence of theorems and taking numerical errors into account. Postselection aims to increase the secure key rate by removing those parts of the key, where a potential adversary might have gained more information than the communicating parties. The investigations are carried out both for the untrusted ideal and the trusted non-ideal detector scenario and we provide novel analytical results for the operators related to the postselection map.For four-state protocols, we demonstrate that a new cross-shaped postselectionstrategy out performs the state-of-the-art radial postselection scheme clearly in regions of medium to high transmission distances and medium to high values of noise and performs comparable to a more complicated radial&angular scheme. As the error-correction phase is a known bottleneck for many real QKD systems, we examined the secure key rate when a large fraction of the raw key is removed by postselection. We observe that a smart choice of the postselection strategy can increase the key rate while lowering the raw key rate, hence the computational effort in the error-correction phase. One can think this even further: For cross-shaped postselection we showed that the secure key rate is roughly 80% of the secure keyrate without postselection when only 20% of the raw key passes the postselection.For high noise levels this can be increased even further. The cross-shaped post-selection strategy can easily be implemented in the data processing of both new and existing CV-QKD systems, hence can increase the achievable secure key rate significantly. Furthermore, we examined the radial&angular postselection strategy which combines the advantages of the radial postselection scheme for low transmission distances and those of the cross-shaped postselection scheme for medium and high transmission distances on the cost of higher complexity.Additionally, we examine an eight-state phase-shift keying protocol and compare the obtained key rates to the key rates for the four-state protocol and investigate a radial postselection scheme. We observe that the 8PSK protocol yields higher secure key rates and reaches higher maximal achievable transmission distances than a QPSK protocol with a comparable postselection strategy, in particular for medium to high values of excess noise. Furthermore, we explore the relation between the achievable secure key rate and the probability to pass the postselectionphase for various noise-levels and two different practically relevant reconciliation efficiencies. This leads to similar strategies as found for the QPSK protocols to reduce the raw key rate significantly while decreasing the secure key rate only moderately. For very high values of excess noise, the raw key rate can be increased further compared to the key rate achieved without performing postselection whileremoving a large fraction of the raw key rate.