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Physical Review Applied

Quantum proof and high-speed privacy amplification for quantum key distribution

Bing-Qi Yan, Junbin Wu, Fan Fan, Bingjie Xu, Xiaoling Zhang, Min Gu

Published May 2, 2025• 6 citations
Publisher

Abstract

Privacy amplification (PA) is a vital step in the postprocessing stage of quantum key distribution, which can extract a secure key from a source partially leaked to the environment. We propose a PA scheme that uses the square hash (SQH)--modular arithmetic hash (MH) function. In consideration of a potential quantum attacker, we use the trace distance as a metric to measure the closeness between two classical-quantum states. We transform a raw key state to a processed state using this scheme and derive the trace distance between the processed state and the ideal target state. It is proven that the trace distance can be reached as small as desired. Additionally, our approach accelerates the SQH-MH function through the number-theoretic transform, reducing the computational complexity to $O(n\mathrm{log}n)$. On the basis of a CPU, a maximum throughput rate of 1.06 Gbit/s is achieved, and the results show that our algorithm demonstrates good stability of the throughput rate when handling tasks with different compression ratios.

DOI
10.1103/physrevapplied.23.054004

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EPJ Quantum Technology•Dec 1, 2025•1 citation
Privacy amplification (PA) is a vital step in the postprocessing stage of quantum key distribution, which can extract a secure key from a source partially leaked to the environment. We propose a PA scheme that uses the square hash (SQH)--modular arithmetic hash (MH) function. In consideration of a potential quantum attacker, we use the trace distance as a metric to measure the closeness between two classical-quantum states. We transform a raw key state to a processed state using this scheme and derive the trace distance between the processed state and the ideal target state. It is proven that the trace distance can be reached as small as desired. Additionally, our approach accelerates the SQH-MH function through the number-theoretic transform, reducing the computational complexity to O(nlogn)O(n\mathrm{log}n)O(nlogn). On the basis of a CPU, a maximum throughput rate of 1.06 Gbit/s is achieved, and the results show that our algorithm demonstrates good stability of the throughput rate when handling tasks with different compression ratios.
Privacy amplification (PA) is a critical process in quantum key distribution (QKD) systems, essential for eliminating information leakage to eavesdroppers and extracting unconditionally secure keys. In practical high-speed QKD systems, a large input size enables the PA scheme to approach the asymptotic limit of the secure-key rate. Concurrently, the throughput of the PA scheme must keep pace with the GHz-level clock rates of modern QKD systems to prevent it from becoming a systemic bottleneck. To globally optimize these objectives, this paper proposes a scalable and highly efficient PA scheme combining a novel δ-almost-universal hash family (FM3H) with modular arithmetic hashing (MH), and proves the security of this hybrid scheme, within the information-theoretic and quantum-side framework. The proposed scheme achieves a throughput of 283 Mbps on a common mobile CPU with an input size of 1010 bits. NIST statistical tests and avalanche-effect measurements are reported as characterizations of the statistical properties of the output sequences. To our knowledge, this work presents the first FPGA-based implementation of a modular hashing-based PA scheme, thereby breaking the dependence of modular hashing schemes on general software platforms. In DV-QKD systems, the FPGA-based scheme achieves a throughput of 1.706 Gbps while maintaining low resource overhead.
Privacy amplification (PA) is an essential part in a quantum key distribution (QKD) system, distilling a highly secure key from a partially secure string by public negotiation between two parties. The optimization objectives of privacy amplification for QKD are large block size, high throughput and low cost. For the global optimization of these objectives, a novel privacy amplification algorithm is proposed in this paper by combining multilinear-modular-hashing and modular arithmetic hashing. This paper proves the security of this hybrid hashing PA algorithm within the framework of both information theory and composition security theory. A scheme based on this algorithm is implemented and evaluated on a CPU platform. The results on a typical CV-QKD system indicate that the throughput of this scheme (261Mbps@2.6*10^8 input block size) is twice higher than the best existing scheme (140Mbps@1*10^8 input block size). Moreover, This scheme is implemented on a mobile CPU platform instead of a desktop CPU or a server CPU, which means that this algorithm has a better performance with a much lower cost and power consumption.
ABSTRACT Privacy amplification (PA) is a crucial step in Quantum Key Distribution (QKD). The original key with the risk of information leakage is transformed into the final key with information theory security through PA. The speed of PA limits the overall rate of QKD. This article proposes a novel approach of PA. This method connects Square Hash (SQH) and number‐theory transform Hash (NH) in series and utilizes number‐theoretic fast Fourier transform (NTT) to reduce its computational complexity to . This paper provides a security proof based on information theory. To implement the scheme based on field programmable gate arrays (FPGA), this paper designed a new architecture to support it. Several optimization techniques of the PA scheme have been implemented to accelerate computation speed and reduce resource consumption, including parallel NTT computation unit and pipelined carry unit, etc. This architecture is also applicable to other similar PA scheme. The experiment result achieves a higher throughput than the compared FPGA‐based works. After further optimization of the architecture, the clock frequency reaches 250 MHz, and the throughput exceeds 10 Gbps, achieving more than 6 times higher than the fastest PA solution among the compared works.
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.
Privacy amplification (PA) is the art of distilling a highly secret key from a partially secure string by public discussion. It is a vital procedure in quantum key distribution (QKD) to produce a theoretically unconditional secure key. The throughput of PA has become the bottleneck of most high-speed discrete variable QKD (DV-QKD) systems. Although some Toeplitz-hash PA schemes can meet the demand of throughput, their high throughput extremely depends on the high cost platform, such as MIC or GPU. From the comprehensive view of development cost, integration level and power consumption, CPU is a general low cost platform. However, the throughput of CPU based PA scheme is not satisfactory so far, mainly due to the conflict between the intrinsic serial characteristic of CPU and the parallel requirement of high throughput Toeplitz-hash PA scheme. In this paper, a high throughput modular arithmetic hash PA scheme using GNU multiple precision arithmetic library (GMP) based on CPU platform is proposed. The experimental results show that the throughput of our scheme is nearly an order of magnitude higher than the comparative scheme on the similar CPU platform, which is 135 Mbps and 69 Mbps at the block sizes of 106 and 108 on Intel i3-2120 CPU respectively. Moreover, our scheme can provide the best throughput among DV-QKD PA schemes, which is 260 Mbps and 140 Mbps at the block sizes of 106and 108on Intel i9-9900k CPU respectively.
Abstract Privacy Amplification (PA) is indispensable in Quantum Key Distribution (QKD), ensuring security against eavesdropping by eliminating information leakage. For Discrete-Variable QKD (DV-QKD) protocols, a large input block size exceeding 108 bits is preferred to achieve the secure key rate approaching the asymptotic limit. However, in state-of-the-art quantum key distribution systems operating at multi-GHz pulse rates, PA becomes a critical bottleneck due to the conflicting requirements of large input block sizes and high throughput. To address this challenge, we propose a novel PA algorithm utilizing a newly constructed universal hash family DM3H and prove its cryptographic security rigorously. Based on the PA algorithm, we design and implement an efficient PA scheme which is capable of processing input block sizes up to 10 10 bits while achieving high throughput performance. For an input block size of 1010 bits, the implementation on the i9-14900 platform demonstrates a throughput of 112 Mbps with a retention ratio of 0.33. This breakthrough significantly enhances the secure key rate and maximum transmission distance of DV-QKD systems.