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IEEE Access

Arithmetic Reconciliation for CVQKD With Gaussian Modulation

Rávilla R. S. Leite, Juliana Martins de Assis, Micael Andrade Dias, Francisco de Assis

Published Jan 1, 2026• 0 citations
Publisher

Abstract

We introduce a new and applicable technique for reconciling continuous-variable quantum key distribution (CVQKD) based on Gaussian modulation of coherent states. Its key feature is the application of the cumulative distribution function (CDF) to map Gaussian variables onto uniform $[{0,1}]$ random variables, which are then expanded into Bernoulli $(1/2)$ random variables, following the principles of Arithmetic Coding. This approach replaces the complex quantization procedures inherent in Slice Reconciliation, while also normalizing the variables of interest to generate the secret key, and thus avoiding the complexities associated with Multidimensional Reconciliation algorithms. The protocol does not use any post-selection, thus it can be proved secure against arbitrary collective attacks.

DOI
10.1109/access.2026.3738025

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Feb 19, 2014•20 citations
We introduce a new and applicable technique for reconciling continuous-variable quantum key distribution (CVQKD) based on Gaussian modulation of coherent states. Its key feature is the application of the cumulative distribution function (CDF) to map Gaussian variables onto uniform [0,1][{0,1}][0,1] random variables, which are then expanded into Bernoulli (1/2)(1/2)(1/2) random variables, following the principles of Arithmetic Coding. This approach replaces the complex quantization procedures inherent in Slice Reconciliation, while also normalizing the variables of interest to generate the secret key, and thus avoiding the complexities associated with Multidimensional Reconciliation algorithms. The protocol does not use any post-selection, thus it can be proved secure against arbitrary collective attacks.
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.
We propose a method for extracting an errorless secret key in a continuous-variable quantum key distribution protocol, which is based on Gaussian modulation of coherent states and homodyne detection. The crucial feature is an eight-dimensional reconciliation method based on the algebraic properties of octonions. Since the protocol does not use any post-selection, it can be proven secure against arbitrary collective attacks by using well-established theorems on the optimality of Gaussian attacks. By using this coding scheme with an appropriate signal-to-noise ratio, the distance for a secure continuous-variable quantum key distribution can be significantly extended.
The two-way continuous-variable quantum key distribution (CVQKD) systems allow higher key rates and improved transmission distances over standard telecommunication networks in comparison to the one-way CVQKD protocols. To exploit the real potential of two-way CVQKD systems a robust reconciliation technique is needed. It is currently unavailable, which makes it impossible to reach the real performance of a two-way CVQKD system. The reconciliation process of correlated Gaussian variables is a complex problem that requires either tomography in the physical layer that is intractable in a practical scenario, or high-cost calculations in the multidimensional spherical space with strict dimensional limitations. To avoid these issues, we propose an efficient logical layer-based reconciliation method for two-way CVQKD to extract binary information from correlated Gaussian variables. We demonstrate that by operating on the raw-data level, the noise of the quantum channel can be corrected in the scalar space and the reconciliation can be extended to arbitrary high dimensions. We prove that the error probability of scalar reconciliation is zero in any practical CVQKD scenario, and provides unconditional security. The results allow to significantly improve the currently available key rates and transmission distances of two-way CVQKD. The proposed scalar reconciliation can also be applied in one-way systems as well, to replace the existing reconciliation schemes.
Continuous variable quantum key distribution allows two legitimate parties to share a common secret key and encompasses reconciliation protocols. A relatively new reconciliation protocol, Arithmetic Reconciliation, presents low complexity and has increasing reconciliation efficiency with lower SNRs. In this paper, we obtain reconciliation efficiencies for this protocol in realistic scenarios, by means of estimation of mutual information, and we also present rates for sequence match of secret keys by Alice and Bob. Results show that this technique is feasible and promising to continuous variable quantum key distribution applications.
We propose an efficient logical layer-based reconciliation method for continuous-variable quantum key distribution (CVQKD) to extract binary information from correlated Gaussian variables. We demonstrate that by operating on the raw-data level, the noise of the quantum channel can be corrected in the low-dimensional (scalar) space, and the reconciliation can be extended to arbitrary dimensions. The CVQKD systems allow an unconditionally secret communication over standard telecommunication networks. To exploit the real potential of CVQKD a robust reconciliation technique is needed. It is currently unavailable, which makes it impossible to reach the real performance of the CVQKD protocols. The reconciliation is a post-processing step separated from the transmission of quantum states, which is aimed to derive the secret key from the raw data. The reconciliation process of correlated Gaussian variables is a complex problem that requires either tomography in the physical layer that is intractable in a practical scenario, or high-cost calculations in the multidimensional spherical space with strict dimensional limitations. To avoid these issues, we define the low-dimensional reconciliation. We prove that the error probability of one-dimensional reconciliation is zero in any practical CVQKD scenario, and provides unconditional security. The results allow for significantly improving the currently available key rates and transmission distances of CVQKD.
The two-way continuous-variable quantum key distribution (CVQKD) systems allow higher key rates and improved transmission distances over standard telecommunication networks in comparison to the one-way CVQKD protocols. To exploit the real potential of two-way CVQKD systems a robust reconciliation technique is needed. It is currently unavailable, which makes it impossible to reach the real performance of a two-way CVQKD system. The reconciliation process of correlated Gaussian variables is a complex problem that requires either tomography in the physical layer that is intractable in a practical scenario, or high-cost calculations in the multidimensional spherical space with strict dimensional limitations. To avoid these issues, we propose an efficient logical layer-based reconciliation method for two-way CVQKD to extract binary information from correlated Gaussian variables. We demonstrate that by operating on the raw-data level, the noise of the quantum channel can be corrected in the scalar space and the reconciliation can be extended to arbitrary high dimensions. We prove that the error probability of scalar reconciliation is zero in any practical CVQKD scenario, and provides unconditional security. The results allow to significantly improve the currently available key rates and transmission distances of two-way CVQKD. The proposed scalar reconciliation can also be applied in oneway systems as well, to replace the existing reconciliation schemes.