Bennett-Brassard 1984 Protocol Research Articles

Security proof for single-photon round-robin differential-quadrature-phase-shift quantum key distribution

Combining the differential-quadrature-phase-shift (DQPS) quantum key distribution (QKD) protocol and the recently proposed round-robin-differential-phase-shift (RRDPS), the round-robin-differential-quadrature-phase-shift (RRDQPS) protocol has been proposed. However, a security proof of the RRDQPS QKD protocol is still missing and its advantage over the original RRDPS protocol is also unknown. Here, we give the security proof of single-photon RRDQPS protocol, by which we clearly show that, if a single photon source is applied, RRDQPS protocol will overwhelm the original RRDPS protocol in terms of tolerable error rate. More interestingly, our proof gives a unified theory covering the well-known Bennett-Brassard 1984 (BB84) protocol and RRDQPS protocol. Thus, it is confirmed that BB84 protocol is indeed a special case of the RRDQPS protocol.

Evaluation of the phase randomness of a light source in quantum-key-distribution systems with an attenuated laser

The phase randomized light is one of the key assumptions in the security proof of Bennett-Brassard 1984 (BB84) quantum key distribution (QKD) protocol implemented with an attenuated laser. Though the assumption has been believed to be satisfied for conventional systems, it should be reexamined for current high speed QKD systems. The phase correlation may be induced by the overlap of the optical pulses, the interval of which decreases as the clock frequency. The phase randomness was investigated experimentally by measuring the visibility of interference. An asymmetric Mach-Zehnder interferometer was used to observe the interference between adjacent pulses from a gain-switched distributed feedback laser diode driven at 10 GHz. Low visibility was observed when the minimum drive current was set far below the threshold, while the interference emerged when the minimum drive current was close to the threshold. Theoretical evaluation on the impact of the imperfect phase randomization provides target values for the visibility to guarantee the phase randomness. The experimental and theoretical results show that secure implementation of decoy BB84 protocol is achievable even for the 10-GHz clock frequency, by using the laser diode under proper operating conditions.

Effects of preparation and measurement misalignments on the security of the Bennett-Brassard 1984 quantum-key-distribution protocol

The ideal Bennett-Brassard 1984 (BB84) quantum key distribution protocol is based on the preparation and measurement of qubits in two alternative bases differing by an angle of pi/2. Any real implementation of the protocol, though, will inevitably introduce misalignments in the preparation of the states and in the alignment of the measurement bases with respect to this ideal situation. Various security proofs take into account (at least partially) such errors, i.e., show how Alice and Bob can still distil a secure key in the presence of these imperfections. Here, we consider the complementary problem: how can Eve exploit misalignments to obtain more information about the key than would be possible in an ideal implementation? Specifically, we investigate the effects of misalignment errors on the security of the BB84 protocol in the case of individual attacks, where necessary and sufficient conditions for security are known. Though the effects of these errors are small for expected deviations from the perfect situation, our results nevertheless show that Alice and Bob can incorrectly conclude that they have established a secure key if the inevitable experimental errors in the state preparation and in the alignment of the measurements are not taken into account. This gives further weight to the idea that the formulation and security analysis of any quantum cryptography protocol should be based on realistic assumptions about the properties of the apparatus used. Additionally, we note that BB84 seems more robust against alignment imperfections if both the x and z bases are used to generate the key.

Robust variations of the Bennett-Brassard 1984 protocol against collective noise

The security of the decoherence-free version of the Bennett-Brassard 1984 (BB84) protocol [A. Cabello, Phys. Rev. A 75, 020301 (2007)] is analyzed and shown to be vulnerable under the intercept-resend attack. We propose two improved versions of this protocol. Both improvements remain the performance of robustness against collective noise and refuse the security flaw. Especially, the second improvement, which is called four-qubit decoherence-free (DF) BB84 protocol, not only remains all characteristics of the original protocol but also has a higher efficiency. We also give a detailed security proof of four-qubit DF BB84 protocol.

Quantum Kolmogorov complexity and quantum key distribution

We discuss the Bennett-Brassard 1984 (BB84) quantum key distribution protocol in the light of quantum algorithmic information. While Shannon's information theory needs a probability to define a notion of information, algorithmic information theory does not need it and can assign a notion of information to an individual object. The program length necessary to describe an object, Kolmogorov complexity, plays the most fundamental role in the theory. In the context of algorithmic information theory, we formulate a security criterion for the quantum key distribution by using the quantum Kolmogorov complexity that was recently defined by Vit\'anyi. We show that a simple BB84 protocol indeed distributes a binary sequence between Alice and Bob that looks almost random for Eve with a probability exponentially close to 1.

Tomography increases key rates of quantum-key-distribution protocols

We construct a practically implementable classical processing for the Bennett-Brassard 1984 (BB84) protocol and the six-state protocol that fully utilizes the accurate channel estimation method, which is also known as the quantum tomography. Our proposed processing yields at least as high a key rate as the standard processing by Shor and Preskill. We show two examples of quantum channels over which the key rate of our proposed processing is strictly higher than the standard processing. In the second example, the BB84 protocol with our proposed processing yields a positive key rate even though the so-called error rate is higher than the 25% limit.

Security proof for quantum-key-distribution systems with threshold detectors

In this paper, we rigorously prove the intuition that in security proofs for the Bennett-Brassard 1984 (BB84) protocol, one may regard an incoming signal to Bob as a qubit state. From this result, it follows that all security proofs for BB84 protocol based on a virtual qubit entanglement distillation protocol, which was originally proposed by Lo and Chau [Science 283, 2050 (1999)] and by Shor and Preskill [Phys. Rev. Lett. 85, 441 (2000)], are all valid even if Bob's actual apparatus cannot distill a qubit state explicitly. As a consequence, especially, the well-known result that a higher bit error rate of 20% can be tolerated for BB84 protocol by using two-way classical communications is still valid even when Bob uses threshold detectors. Using the same technique, we also prove the security of Bennett-Brassard-Mermin 1992 (BBM92) protocol where Alice and Bob both use threshold detectors.

Squashing Models for Optical Measurements in Quantum Communication

Measurements with photodetectors are naturally described in the infinite dimensional Fock space of one or several modes. For some measurements, a model has been postulated which describes the full measurement as a composition of a mapping (squashing) of the signal into a small dimensional Hilbert space followed by a specified target measurement. We present a formalism to investigate whether a given measurement pair of full and target measurements can be connected by a squashing model. We show that a measurement used in the Bennett-Brassard 1984 (BB84) protocol does allow a squashing description, although the corresponding six-state protocol measurement does not. As a result, security proofs for the BB84 protocol can be based on the assumption that the eavesdropper forwards at most one photon, while the same does not hold for the six-state protocol.

Efficiently implementable codes for quantum key expansion

The Shor-Preskill proof of the security of the Bennett-Brassard 1984 (BB84) quantum key distribution protocol relies on the theoretical existence of good classical error-correcting codes with the ``dual-containing'' property. A practical implementation of the BB84 protocol thus requires explicit and efficiently decodable constructions of such codes, which are not known. On the other hand, modern coding theory abounds with non-dual-containing codes with excellent performance and efficient decoding algorithms. We show that the dual-containing constraint can be lifted at a small price: instead of a key distribution protocol, an efficiently implementable key expansion protocol is obtained, capable of increasing the size of a preshared key by a constant factor.

Security proof of a three-state quantum-key-distribution protocol without rotational symmetry

Standard security proofs of quantum-key-distribution (QKD) protocols often rely on symmetry arguments. In this paper, we prove the security of a three-state protocol that does not possess rotational symmetry. The three-state QKD protocol we consider involves three qubit states, where the first two states $\ensuremath{\mid}{0}_{z}⟩$ and $\ensuremath{\mid}{1}_{z}⟩$ can contribute to key generation, and the third state $\ensuremath{\mid}+⟩=(\ensuremath{\mid}{0}_{z}⟩+\ensuremath{\mid}{1}_{z}⟩)∕\sqrt{2}$ is for channel estimation. This protocol has been proposed and implemented experimentally in some frequency-based QKD systems where the three states can be prepared easily. Thus, by founding on the security of this three-state protocol, we prove that these QKD schemes are, in fact, unconditionally secure against any attacks allowed by quantum mechanics. The main task in our proof is to upper bound the phase error rate of the qubits given the bit error rates observed. Unconditional security can then be proved not only for the ideal case of a single-photon source and perfect detectors, but also for the realistic case of a phase-randomized weak coherent light source and imperfect threshold detectors. Our result in the phase error rate upper bound is independent of the loss in the channel. Also, we compare the three-state protocol with the Bennett-Brassard 1984 (BB84) protocol. For the single-photon source case, our result proves that the BB84 protocol strictly tolerates a higher quantum bit error rate than the three-state protocol, while for the coherent-source case, the BB84 protocol achieves a higher key generation rate and secure distance than the three-state protocol when a decoy-state method is used.

Effects of detector efficiency mismatch on security of quantum cryptosystems

We suggest a type of attack on quantum cryptosystems that exploits variations in detector efficiency as a function of a control parameter accessible to an eavesdropper. With gated single-photon detectors, this control parameter can be the timing of the incoming pulse. When the eavesdropper sends short pulses using the appropriate timing so that the two gated detectors in Bob's setup have different efficiencies, the security of quantum key distribution can be compromised. Specifically, we show for the Bennett-Brassard 1984 (BB84) protocol that if the efficiency mismatch between 0 and 1 detectors for some value of the control parameter gets large enough (roughly 15:1 or larger), Eve can construct a successful faked-states attack causing a quantum bit error rate lower than 11%. We also derive a general security bound as a function of the detector sensitivity mismatch for the BB84 protocol. Experimental data for two different detectors are presented, and protection measures against this attack are discussed.

Unconditionally secure key distillation from multiphotons

In this paper, we prove that the unconditionally secure key can be surprisingly extracted from multiphoton emission part in the photon polarization-based quantum key distribution. One example is shown by explicitly proving that one can indeed generate an unconditionally secure key from Alice's two-photon emission part proposed by Scarani [ et al. Phys. Rev. Lett. 92, 057901 (2004)]. Which is called the Scarani-Acin-Ribordy-Gisin (SARG04) protocol. This protocol uses the same four states as in Bennett-Brassard 1984 (BB84) and differs only in the classical postprocessing protocol. It is, thus, interesting to see how the classical postprocessing of quantum key distribution might qualitatively change its security. We also show that one can generate an unconditionally secure key from the single to the four-photon part in a generalized SARG04 protocol that uses six states. Finally, we also compare the bit error rate threshold of these protocols with the one in the BB84 protocol and the original six-state protocol assuming a depolarizing channel.

Conclusive eavesdropping in quantum key distribution

It is argued that the BB84 (Bennett–Brassard 1984) protocol of quantum key distribution has a vulnerability similar to the well-known vulnerability of the B92 (Bennett 1992) protocol in the presence of losses. The quantum circuit and design are first reviewed for an optimized entangling probe attacking the BB84 protocol of quantum key distribution and yielding maximum information to the probe. Probe photon polarization states become optimally entangled with the signal states on their way between the legitimate transmitter and receiver. Although standard von Neumann projective measurements of the probe yield maximum information on the pre-privacy amplified key, if instead the probe measurements are performed with a certain positive operator valued measure, then the measurement results are unambiguous, at least some of the time, resulting in complete information gain by the probe for sufficient loss in the key distribution channel.

Unambiguous State Discrimination in Quantum Key Distribution

The quantum circuit and design are presented for an optimized entangling probe attacking the BB84 Protocol of quantum key distribution (QKD) and yielding maximum information to the probe. Probe photon polarization states become optimally entangled with the signal states on their way between the legitimate transmitter and receiver. Although standard von-Neumann projective measurements of the probe yield maximum information on the pre-privacy amplified key, if instead the probe measurements are performed with a certain positive operator valued measure (POVM), then the measurement results are unambiguous, at least some of the time. It follows that the BB84 (Bennett---Brassard 1984) protocol of quantum key distribution has a vulnerability similar to the well-known vulnerability of the B92 (Bennett 1992) protocol

Reliability of Calderbank–Shor–Steane codes and security of quantum key distribution

After Mayers (1996, 2001) gave a proof of the security of the Bennett-Brassard 1984 (BB84) quantum key distribution protocol, Shor and Preskill (2000) made a remarkable observation that a Calderbank-Shor-Steane (CSS) code had been implicitly used in the BB84 protocol, and suggested its security could be proven by bounding the fidelity, say F(n), of the incorporated CSS code of length n in the form 1-F(n) <= exp[-n E+o(n)] for some positive number E. This work presents such a number E=E(R) as a function of the rate of a code R, and a threshold R' such that E(R)>0 whenever R < R', which is larger than the achievable rate based on the Gilbert-Varshamov bound that is essentially due to Shor and Preskill (2000). The codes in the present work are robust against fluctuations of channel parameters, which fact is needed to establish the security rigorously and was not proved for rates above the Gilbert-Varshamov rate before in the literature. As a byproduct, the security of a modified BB84 protocol against any joint (coherent) attacks is proved quantitatively.

Shor-Preskill-type security proof for concatenated Bennett-Brassard 1984 quantum-key-distribution protocol

We discuss a long code problem in the Bennett-Brassard 1984 (BB84) quantum-key-distribution protocol and describe how it can be overcome by concatenation of the protocol. Observing that concatenated modified Lo-Chau protocol finally reduces to the concatenated BB84 protocol, we give the unconditional security of the concatenated BB84 protocol.

Simple proof of the unconditional security of the Bennett 1992 quantum key distribution protocol

It is generally accepted that quantum key distribution (QKD) could supply legitimate users with unconditional security during their communication. Quite a lot of satisfactory efforts have been achieved on experimentations with quantum cryptography. However, when the eavesdropper has extra-powerful computational ability, has access to a quantum computer, for example, and can carry into execution any eavesdropping measurement that is allowed by the laws of physics, the security against such attacks has not been widely studied and rigorously proved for most QKD protocols. Quite recently, Shor and Preskill proved concisely the unconditional security of the Bennett-Brassard 1984 (BB84) protocol. Their method is highly valued for its clarity of concept and concision of form. In order to take advantage of the Shor-Preskill technique in their proof of the unconditional security of the BB84 QKD protocol, we introduced in this paper a transformation that can translate the Bennett 1992 (B92) protocol into the BB84 protocol. By proving that the transformation leaks no more information to the eavesdropper, we proved the unconditional security of the B92 protocol. We also settled the problem proposed by Lo about how to prove the unconditional security of the B92 protocol with the Shor-Preskill method.