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IEEE photonics journal

High-Speed Privacy Amplification Algorithm Based on Two-Dimensional Cellular Automata in Quantum Key Distribution

Encheng Tian, Han Hai, Xue-Qin Jiang, Enjian Bai, Genlong Chen, Peng Huang, Guihua Zeng

Published Dec 8, 2025• 2 citations
Publisher

Abstract

Privacy amplification is a critical component in quantum key distribution (QKD) to eliminate eavesdropper information and distill unconditionally secure keys. While conventional Toeplitz hash privacy amplification algorithms face challenges in computational complexity and hardware resource demands, existing one-dimensional cellular automata (1D CAs) algorithm lacks sufficient parallelism and exhibits limited diffusion under finite-size constraints. To overcome these limitations, a privacy amplification algorithm using two-dimensional cellular automata (2D CAs) for QKD is presented in this paper. The proposed algorithm decreases the computation complexity and increases the processing speed. Unlike conventional Toeplitz hash algorithms, the proposed algorithm utilizes the inherent parallelism of 2D CAs to enable simultaneous multi-bit confusion through cyclic row shifts and XOR operations. Furthermore, we prove that the 2D CAs-based algorithm is a universal hash family and satisfies the principle of privacy amplification. The randomness of the proposed algorithm was evaluated through the NIST test suite and an avalanche test, both of which indicated great performance. Finally, we implement the proposed algorithm in field-programmable gate array (FPGA). The experimental results on a Xilinx Artix-7 FPGA demonstrate that our scheme achieves high throughput and significantly reduces hardware resource consumption.

DOI
10.1109/jphot.2025.3638669

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Privacy amplification is a critical component in quantum key distribution (QKD) to eliminate eavesdropper information and distill unconditionally secure keys. While conventional Toeplitz hash privacy amplification algorithms face challenges in computational complexity and hardware resource demands, existing one-dimensional cellular automata (1D CAs) algorithm lacks sufficient parallelism and exhibits limited diffusion under finite-size constraints. To overcome these limitations, a privacy amplification algorithm using two-dimensional cellular automata (2D CAs) for QKD is presented in this paper. The proposed algorithm decreases the computation complexity and increases the processing speed. Unlike conventional Toeplitz hash algorithms, the proposed algorithm utilizes the inherent parallelism of 2D CAs to enable simultaneous multi-bit confusion through cyclic row shifts and XOR operations. Furthermore, we prove that the 2D CAs-based algorithm is a universal hash family and satisfies the principle of privacy amplification. The randomness of the proposed algorithm was evaluated through the NIST test suite and an avalanche test, both of which indicated great performance. Finally, we implement the proposed algorithm in field-programmable gate array (FPGA). The experimental results on a Xilinx Artix-7 FPGA demonstrate that our scheme achieves high throughput and significantly reduces hardware resource consumption.
Privacy amplification (PA) is a vital procedure in quantum key distribution (QKD) to shrink the eavesdropper's information about the final key almost to zero. With the increase of repeat frequency of discrete variable QKD (DV-QKD) system, PA processing speed has become the bottleneck in many high-speed DV-QKD systems. In this paper, a high-speed adaptive field-programmable gate array (FPGA)-based PA scheme using a fast Fourier transform (FFT) is presented. To decrease the computation complexity, a modified 2-D FFT-based Toeplitz PA scheme is designed. To increase the processing speed of the scheme on the constraint of limited resources, a real-value oriented FFT acceleration method and a fast read/write balanced matrix transposition method are designed and implemented in our scheme. The experimental results on a Xilinx Virtex-6 FPGA demonstrate that the throughput is nearly double of the latest FPGA based Toeplitz PA scheme according to the literature. Besides, this scheme owns not only the good adaptivity to compression ratio but also the compression ratio independent resource consumption. Therefore, this scheme can fit many high-speed QKD applications.
Privacy amplification (PA) is a vital procedure in quantum key distribution (QKD) to generate the secret key that the eavesdropper has only negligible information from the identical correcting key for the communicating parties. With the increase of repeat frequency of discrete-variable QKD (DV-QKD) system, the processing speed of PA has become the bottle neck restricting DV-QKD's secure key rate. The PA using Toeplitz-based Hash function is adopted widely because of its simplicity and parallel feature. Because this algorithm can be accelerated with Fast Fourier Transform (FFT), an improved scheme PA for Field-programmable Gate Array (FPGA) based on this is proposed. This paper improves the custom FFT-based algorithm by reducing the number of computations and read/write memory operations significantly. The correctness is verified when implemented in a Xilinx Virtex-6 FPGA. Meanwhile, the processing speed of improved scheme can nearly double the classical Toeplitz Hashing scheme on FPGA through the actual experiment.
For a high-speed and secure continuous-variable quantum key distribution (CV-QKD) system, privacy amplification (PA) plays an important role. To reduce the finite size effect, the input length of PA should be at least on the order of 10^8, 10^9, 10^10 when the transmission distance is about 50km, 80km, 100km, respectively. This leads to high computation complexity and large storage demand of the data, which is unfriendly to field programmable gate array (FPGA) implementation for its limited resource. In addition, the limited IO speed of Double Data Rate Synchronous Dynamic Random Access Memory (DDR SDRAM) restricts the implementation performance of PA. In this paper, we propose an effective way to access data based on DDR to improve the performance of PA. As a result, the reading time from DDR can be reduced, and it can eliminate the effect of the limited IO speed of DDR, so that PA can perform with multiple code-words. This can make full use of the resource of FPGA and increase the execution speed of PA. Besides, combining with the proposed method, an easier algorithm is used to decrease the complexity of calculations. Based on these methods, we realize PA with Toeplitz matrix based on FPGA and the experimental throughput is about 288Mbps when the input length is about 100Mbits.
Privacy amplification is an important step in the post-processing of quantum communication, which plays an indispensable role in the security of quantum key distribution systems. In this paper, we propose a Cellular Automata-based privacy amplification algorithm, which improves the speed of key distribution. The proposed algorithm is characterized by block iteration to generate secure key of arbitrary length. The core of the algorithm in this paper is to use the property that Cellular Automata can generate multiple new associated random sequences at the same time to carry out bit operations for multiple negotiation keys in the meantime and calculate in turn, so as to quickly realize the compression of negotiation keys. By analyzing the final key, the proposed algorithm has the advantages of fast key generation speed and high real-time performance. At the same time, the results of the NIST randomness test and avalanche test show that the algorithm has good randomness performance.
In a quantum key distribution (QKD) system, privacy amplification (PA) is an essential procedure that can effectively eliminate the leaked information to an eavesdropper and distill a secret key. The processing speed of the PA algorithm inevitably affects the final key rate of the QKD system. We propose a high-speed PA algorithm based on field-programmable gate array (FPGA), where the matrix multiplications are divided into a number of rhomboid-block operations. The implementation of the PA reduces significantly the required number of FPGA Block RAMs. Meanwhile, it automatically adapts to different lengths of input and output blocks. Finally, we verified the correctness and high-speed of the algorithm when implemented in a Xilinx Virtex-7 FPGA.
In this paper, a high-performance field-programmable gate array (FPGA) based privacy amplification (PA) scheme for both continuous variable and discrete variable - quantum key distribution (QKD) is presented, which can provide more than 1.5 Gbps throughput at the input block size close to10810^{8}108. Besides, the resource consumption is so economical that there is no need of any external storage. To the best of our knowledge, our scheme outperforms existing FPGA based PA schemes by over an order of magnitude in terms of throughput and input block size. To achieve this feature, we design a novel FPGA-based hybrid hash PA algorithm combining multilinear modular hash and modular arithmetic hash (MMH-MH) based on number theoretical transform (NTT). Based on the hybrid hash algorithm, many optimization strategies are designed in our FPGA based PA scheme, such as an NTT-based optimization unit, an NTT-reusable architecture, a memory-saving modular accumulation unit for multilinear modular hash and a pipelined binary modulo unit for modular arithmetic hash. Plentiful experiment results demonstrate the high performance and reliability of our scheme. It can be embedded in most standalone QKD devices conveniently. What's more, it is also a good candidate for a chip-based QKD system.