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Journal of Communication and Information Systems

Enhancing Continuous-Variable Quantum Key Distribution by State Preparation With Shannon-Kotel'nikov Maps

Edmar Nascimento, Francisco de Assis

Published Jan 1, 2019• 0 citations
Publisher

Abstract

We propose a continuous-variable quantum key distribution protocol that uses Shannon-Kotel'nikov maps for preparing coherent quantum states. Our protocol has similarities with the no-switching protocol in the sense that it requires that both quadratures need to be measured. It is also a Gaussian modulated protocol, because the secret key is to be extracted from Gaussian parameters. The use of this kind of map in the preparation of coherent states allows the increase in the source-to-distortion ratio (SDR) between Alice and Bob, thus making information reconciliation easier. This increase in SDR is due to the use of nonlinear maps of higher dimension instead of simply raising Alice's variance. We analyze here two kind of maps: the uniform Archimedes' spiral and the geodesics on a flat torus. We assess the security of our protocol through simulations. In order to do that, we simulate the optimal feedforward attack together with a maximum-likelihood receiver, then we use Kraskov's first algorithm to estimate the mutual information.

DOI
10.14209/jcis.2019.11

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We propose a continuous-variable quantum key distribution protocol that uses Shannon-Kotel'nikov maps for preparing coherent quantum states. Our protocol has similarities with the no-switching protocol in the sense that it requires that both quadratures need to be measured. It is also a Gaussian modulated protocol, because the secret key is to be extracted from Gaussian parameters. The use of this kind of map in the preparation of coherent states allows the increase in the source-to-distortion ratio (SDR) between Alice and Bob, thus making information reconciliation easier. This increase in SDR is due to the use of nonlinear maps of higher dimension instead of simply raising Alice's variance. We analyze here two kind of maps: the uniform Archimedes' spiral and the geodesics on a flat torus. We assess the security of our protocol through simulations. In order to do that, we simulate the optimal feedforward attack together with a maximum-likelihood receiver, then we use Kraskov's first algorithm to estimate the mutual information.
We propose a continuous-variable quantum key distribution (CVQKD) protocol that uses a Shannon- Kotel'nikov expansion map for preparing coherent states. This kind of mapping acts like an analog error correcting code that improves the fidelity between the source signal and the reconstructed one at the receiver. It is well known that a limitation for extending CVQKD to larger distances is the reconciliation efficiency. Such lack of efficiency in reconciliation is mainly due to the difficulty of finding good error correcting codes for low signal-to-noise ratios. Having this limitation in mind, our protocol allows increasing the signal-to-noise ratio between Alice and Bob, thus making reconciliation easier.
Security of continuous-variable quantum key distribution with four-state protocol based on discrete modulation of noisy coherent states is analyzed. Combing discrete modulation and reverse reconciliation, this protocol can be used for long distance cryptography. There is a small difference between the state Alice sends in the discrete modulation protocol and the Gaussian modulation protocol, and it can be treated as excess noise and loss in the channel. As Alice cannot do a precise modulation, she will induce noise to the coherent state. We look this noise as the source noise and derive a lower bound to the secure key rate assuming the eavesdropper cannot benefit from the noise in the source. For avoiding the fast and random phase locking between the signal and local oscillator in experiment, we also analyze the security of four-state protocol using no-switching scheme.
We designed high-efficiency error correcting codes allowing us to extract an errorless secret key in a continuous-variable quantum key distribution (CVQKD) protocol using a Gaussian modulation of coherent states and a homodyne detection. These codes are available for a wide range of signal-to-noise ratios on an additive white Gaussian noise channel with a binary modulation and can be combined with a multidimensional reconciliation method proven secure against arbitrary collective attacks. This improved reconciliation procedure considerably extends the secure range of a CVQKD with a Gaussian modulation, giving a secret key rate of about ${10}^{\ensuremath{-}3}$ bit per pulse at a distance of 120 km for reasonable physical parameters.
A novel quantum key distribution (QKD) protocol is proposed and demonstrated. It is based on the transmission of Gaussian-modulated coherent states that are laser pulses containing a few hundred photons .Shot-noise limited coherent (homodyne) detection is used, which makes it possible to achieve high secret-key rates. It remains, in principle, efficient for any value of the line transmission by use of a "reverse reconciliation" technique, which is shown to be secure against individual attacks based on entanglement and quantum memories. The basic idea is to use a reverse reconciliation protocol, that is, Alice attempts to guess what was received by Bob rather than Bob guessing what was sent by Alice. A reverse protocol always gives Alice an advantage over a potential eavesdropper Eve, regardless the line loss. In addition, a specific advantage of using quantum continuous variables is that the high dimensionality of the phase space may be exploited by modulating the field quadratures with a large dynamics, allowing the encoding of several key bits per pulse. This, together with the fact that fast modulation and detection can be achieved, results in a high-rate secret key distribution.
We discuss quantum key distribution protocols using quantum continuous variables. We show that such protocols can be made secure against individual gaussian attacks regardless the transmission of the optical line between Alice and Bob. %while other ones require that the line transmission is larger than 50%. This is achieved by reversing the reconciliation procedure subsequent to the quantum transmission, that is, using Bob's instead of Alice's data to build the key. Although squeezing or entanglement may be helpful to improve the resistance to noise, they are not required for the protocols to remain secure with high losses. Therefore, these protocols can be implemented very simply by transmitting coherent states and performing homodyne detection. Here, we show that entanglement nevertheless plays a crucial role in the security analysis of coherent state protocols. Every cryptographic protocol based on displaced gaussian states turns out to be equivalent to an entanglement-based protocol, even though no entanglement is actually present. This equivalence even holds in the absence of squeezing, for coherent state protocols. This ``virtual'' entanglement is important to assess the security of these protocols as it provides an upper bound on the mutual information between Alice and Bob if they had used entanglement. The resulting security criteria are compared to the separability criterion for bipartite gaussian variables. It appears that the security thresholds are well within the entanglement region. This supports the idea that coherent state quantum cryptography may be unconditionally secure.
The continuous-variable quantum key distribution (CVQKD) has been demonstrated to be available in practical secure quantum cryptography. However, its performance is restricted strongly by the channel excess noise and the reconciliation efficiency. In this paper, we present a quantum key distribution (QKD) protocol by encoding the secret keys on the random choices of two measurement bases: the conjugate quadratures XXX and PPP. The employed encoding method can dramatically weaken the effects of channel excess noise and reconciliation efficiency on the performance of the QKD protocol. Subsequently, the proposed scheme exhibits the capability to tolerate much higher excess noise and enables us to reach a much longer secure transmission distance even at lower reconciliation efficiency. The proposal can work alternatively to strengthen significantly the performance of the known Gaussian-modulated CVQKD protocol and serve as a multiplier for practical secure quantum cryptography with continuous variables.