Abstract
Stateofart quantum key distribution (QKD) systems are performed with several GHz pulse rates, meanwhile privacy amplification (PA) with large scale inputs has to be performed to generate the final secure keys with quantified security. In this paper, we propose a fast Fourier transform (FFT) enhanced highspeed and largescale (HiLS) PA scheme on commercial CPU platform without increasing dedicated computational devices. The long input weak secure key is divided into many blocks and the random seed for constructing Toeplitz matrix is shuffled to multiple subsequences respectively, then PA procedures are parallel implemented for all subkey blocks with correlated subsequences, afterwards, the outcomes are merged as the final secure key. When the input scale is 128 Mb, our proposed HiLS PA scheme reaches 71.16 Mbps, 54.08 Mbps and 39.15 Mbps with the compression ratio equals to 0.125, 0.25 and 0.375 respectively, resulting achievable secure key generation rates close to the asymptotic limit. HiLS PA scheme can be applied to 10 GHz QKD systems with even larger input scales and the evaluated throughput is around 32.49 Mbps with the compression ratio equals to 0.125 and the input scale of 1 Gb, which is ten times larger than the previous works for QKD systems. Furthermore, with the limited computational resources, the achieved throughput of HiLS PA scheme is 0.44 Mbps with the compression ratio equals to 0.125, when the input scale equals up to 128 Gb. In theory, the PA of the randomness extraction in quantum random number generation (QRNG) is same as the PA procedure in QKD, and our work can also be efficiently performed in highspeed QRNG.
Introduction
Quantum Key Distribution (QKD), which based on the fundamental quantum mechanics, can generate the informationtheoretical secure (ITS) keys for distant communication parties^{1,2,3}. Practical QKD systems are mainly composed of two phases: the quantum communication phase and the postprocessing phase^{4,5}. In the postprocessing phase, partial information about the secure key may still be leaked to the eavesdropper Eve after the key/basis sifting and error correction procedures. Privacy amplification (PA), the most significant postprocessing procedure, coverts the weak secure correlated key to a uniform and ITS key to Eve^{6,7,8}.
Given the input weak secure key W with length of n and the security level ε, the optimal PA scheme in theory can be achieved with (dual) universal hash functions using Toeplitz kind of matrix (T) with computational complexity of O(nlogn)^{9}, and the length of consumed random seed in PA is αn, with minentropy of αn + O(1), α ∈ (0,1]^{10,11,12,13}.
Nowadays, stateofart academic QKD experiments are performed with several GHz pulse rates^{14,15,16,17,18}, advanced multiplexing technologies^{19,20} and extracts secure keys even with highdimensional scenarios^{21,22,23}. Meanwhile, a rigorous statistical fluctuation analysis has to be performed to remove the finitesize key effects on the final secure key^{24,25}. Therefore, a high throughput and largescale (usually larger than several Megabits) PA scheme has to be implemented to realtime extract the secure key with achievable generation rate close to the asymptotic (infinitekey) limit.
The simplest implementation idea of a largescaled PA scheme is directly performing multiplication operation between W and T, resulting in the computational complexity with O(n^{2}). However, such matrixvector multiplication is very suitable to be implemented with FieldProgrammable Gate Array (FPGA) platform. H. Zhang et al. firstly divided T into many smaller blocks and proposed a block parallel PA scheme to speedup the Toeplitz hashing procedure^{26}. S. Yang et al.^{27} and J. Constantin et al.^{28} proposed advanced block partition strategies to reduce the overhead of multiplication operations respectively, resulting in the throughput around 64 Mbps with input scale of 1 megabits^{27}.
Actually, majority optimized PA schemes are performed using fast Fourier transform (FFT) with complexity reduced to O(nlogn)^{8,29,30}. Given fixed security level ε (i.e. 10^{−10}), the farther communication distance, the larger input length of PA scheme should be adapted. For example, in entanglementbased QKD systems, the input length n should be increased to at least the order of 10^{8}. B. Liu et al. firstly improved the throughput of FFT enhanced PA scheme to 60 Mbps with input scale of 12.8 megabits on ManyIntegratedCore (MIC) platform^{8}. Z. L. Yuan et al. implemented a number theoretical transform (NTT) based PA scheme with throughput up to 108.77 Mbps with the input scale of 100 megabits also on MIC platform^{31}. X. Wang et al. proposed a parallel implementation of the lengthcompatible (up to 10 Gbits) FFT based PA algorithm for continuousvariable QKD systems on a graphic processing unit (GPU) platform, with speed over 1 Gbps^{29}.
It’s a huge challenge to implement largescale FFT based PA schemes on FPGA platforms due to the limited resources and ultra complicated hardware design. Implementation of PA schemes on MIC, GPU or other dedicated computational devices consumes ultra high power and volume and significantly increases the design complexity. Improving the throughput of FFT enhanced PA schemes on CPU platforms is a very conventional option, since it can be efficiently integrated to the whole QKD system. However, it’s feasible with CPU implementations only for small input scales (≤10^{6}) and rapidly becomes the performance bottleneck with larger input scales. Therefore, in this article, we propose a fast Fourier transform (FFT) enhanced highspeed and largescale (HiLS) PA scheme on commercial multicore CPU platform. In the HiLS PA scheme, W is divided into many blocks and the random seed for constructing Toeplitz matrix T is shuffled to multiple subsequences respectively, then PA procedures are parallel implemented for all subkey blocks with correlated subsequences, afterwards the outcomes are merged as the final secure key. When the input scale is 128 Mb, our HiLS PA scheme reaches 71.16 Mbps, 54.08 Mbps and 39.15 Mbps with the compression ratio equals to 0.125, 0.25 and 0.375 respectively. Therefore, HiLS PA scheme can be applied to 10 GHz QKD systems with even larger input scales and the evaluated throughput is around 32.49 Mbps with the compression ratio equals to 0.125 and the input scale of 1 Gb, which is ten times larger than the previous works for QKD systems. Furthermore, with the limited computational resources (128 GB memory, 1 TB storage and 16 CPU cores in total), the achieved throughput of HiLS PA scheme is 0.44 Mbps with the compression ratio equals to 0.125, when the input scale equals up to 128 Gb. In theory, the PA of the randomness extraction in quantum random number generation (QRNG) is same as the PA procedure in QKD^{32,33,34}. Thus, HiLS PA scheme can also be efficiently performed in highspeed QRNG.
Related Work
Privacy amplification was first proposed in the context of quantum key distribution by Bennett et al.^{6}, where the channel with perfect authenticity but no privacy (public classical channel) can be used to repair the defects of a channel with imperfect privacy but no authenticity (quantum channel). The schematic diagram of PA in QKD is shown in Fig. 1, Alice and Bob firstly distribute quantum signals via a noisy and lossy quantum channel (fiber or free space), then share correlated and weak secure key W after basis/key sifting and error correction procedures via a public channel. The minentropy of shared weak secure key W is n. Let random variable E summarizes Eve’s entire learned knowledge about W, here, H(WE) ≤ t, t < n. PA, where Alice and Bob publicly discuss a extractor function G:{0,1}^{n}→{0,1}^{r}, such that reduces Eve’s learned information of the final secure key K_{f} from t to at most ε^{6,7,35,36}. Nowadays, most practical extractors are known to the universal hash function, especially the (modified) Toeplitz matrix defined as^{13}
where T(A) is a r × (n − r) Toeplitz matrix, A is a random seed, A = (a_{0}, a_{1}, …, a_{n−1}) ∈ {0,1}^{n−1}, T(A)_{i,j} = a_{j−i+r−1}. Also, we define W_{I} = (w_{0}, w_{1}, …, w_{r−1}) and W_{TA} = (w_{r}, w_{r+1}, …, w_{n−1}). Therefore, the final secure key can be calculated as
In order to efficiently implement the calculation of T(A)W_{TA} using fast Fourier transform (FFT), we have to extend T(A) to a special circulant Toeplitz matrix with scale of (n − 1) × (n − 1) and extend W_{TA} to a vector with length of n − 1 by padding zeros. The optimized multiplication of a circulant matrix and a vector is shown as
where “*” denotes the Hadamard product operator, F denotes the Fourier transform operator, F^{−1} is the inverse Fourier transform operator, X is a vector and H is a circulant Toeplitz matrix with first row h. Since the complexity of F and F^{−1} operations is O(nlogn) and the complexity of Hadmard product operation is O(n), the computational complexity of optimized PA algorithm is O(nlogn)^{8,12}.
In theory, QKD can generate ITS keys for communication parties, even the quantum channel is under control of the eavesdropper Eve. Imperfect implementation and active attacks would leak some information about W to Eve. Alice and Bob can quantify the bound of leaked information accurately with the infinite postprocessing block size. In this paper, we take entanglement based QKD as an example, the secure key rate can be calculated as^{37}
where q is the basis sifting factor, Q_{μ} is the gain of detected entangled photon pairs, ν_{s} is the repetition rate of the entangled source, e_{b} is the measured quantum bit error rate, \({e}_{p}^{U}\) is the estimated upperbound of phase error rate, f(x) is the error correction efficiency, H_{2}(x) is the binary Shannon entropy.
In practice, e_{p}^{U} can not be measured directly and could not be accurately estimated due to the statistical fluctuations with finite postprocessing block sizes. Here, we simulate the required throughput of PA algorithm in a 10 GHz entanglement based QKD with the parameters shown in Table 1. The entangled photon source is put into the middle of communication parties, the finitesizeeffect for the final secure key K_{f} is considered with postprocessing block size from the order of 10^{4} to infinite, and the failure probability ε^{ph} = 10^{−10} for estimating \({e}_{p}^{U}\)^{4}. The analyzed results are shown in Fig. 2, the postprocessing block size should be at least the order of 10^{8} to achieve a secure key rate close to the asymptotic limit. Directly implementing PA algorithms with ultra largescale inputs will limit the performance of full QKD systems. Meanwhile, the required throughput of PA algorithm is around 40 Mbps without any channel loss.
Highspeed and Largescale Privacy Amplification Scheme
The schematic diagram of proposed highspeed and largescale (HiLS) privacy amplification scheme for QKD is shown in Fig. 3. Weak secure key W with length of n is gained after the basis/key sifting and error correction procedures for the measured raw key string at Alice’s (Bob’s) side. Then, Alice and Bob estimate the final secure key length r with rigorous statistical fluctuation analysis procedure. Afterwards, Alice and Bob publicly discuss a random seed with length of n − 1 bits to construct the universal hash function. Our proposed HiLS PA scheme mainly consists of three steps: splitting and shuffling, subPA and securekey merging.
Step 1: Splitting and shuffling
In this step, we divide W to several subvectors and divide the Toeplitz matrix T(A) to submatrices. Assume the scale of submatrix is m × m, m ≤ r. Assume that the Toeplitz matrix T(A) can be divided into t blocks by rows and k blocks by columns, thus in total kt submatrices, \(t=\lceil \frac{nr}{m}\rceil \), \(k=\lceil \frac{r}{m}\rceil \). First of all, we construct a vector A by padding km − r (tm − n + r) zeros to the head (tail) of the exchanged random seed with length of n − 1 bits. Then, we shuffle A into k + t − 1 subvectors, defined as A_{i}: = [a_{im}, a_{im }+_{ 1}, …, a_{(2+i)m−1}], 0 ≤ i < k + t − 1. Therefore, the divided submatrix can be constructed by H_{i,j} = T(A_{i+j}), i ∈ [0, k) and j ∈ [0, t), and we have
where H_{i,j} = H_{i+1, j+1}.
For W, we first pad tm − n + r zeros to the tail and take first r bits and the rest bits to construct the subvector W_{I} and W_{TA}. Then, divide W_{TA} into t subvectors, defined as W_{i}: = [w_{im+r}, w_{im+r+1}, …, w_{(i+1)m+r−1}], where 0 ≤ i < t.
Step 2: SubPA
In this step, the efficient implementation using FFT of multiplication Y_{i,j} is performed to subvector W_{j} and submatrix H_{i,j},
where, i ∈ [0, k) and j ∈ [0, t).
Step 3: SecureKey merging
First, we only take first m bits of Y_{i,j} (defined as \({Y}_{i,j}^{{\rm{a}}}\)), then we merge \({Y}_{i,j}^{{\rm{a}}}\) to vector Z by
Take first r bits of Z (defined as Z^{*}), we can get the final secure key K_{f} by
The detailed implementation of HiLS PA scheme can be described as Algorithm 1. In the procedure of our proposed HiLS PA scheme, we only need to perform k + 2t − 1 times Fourier operations with scale of 2m, kt times Hadamard product operations with scale of m, kt times inverse Fourier operations and kt + 1 times exclusive or (XOR) operations with scale of m. Thus, the computational complexity of the proposed HiLS PA scheme is O(ktmlogm), simplified to around O(nlogm).
Results
The implementation of HiLS PA scheme is evaluated on the multicore server computer, the specifications are shown in Table 2. Due to FFT operation may suffer errors caused by finiteprecision floatpoint arithmetic, we suggest the scale of FFT operation smaller than the order of 10^{8}. Meanwhile, considering the thread synchronization and thread safety issues, the calculations of (inverse) Fourier transforms and also Hadamard products are paralleled in the architecture of shared memory multiprocesses.
We evaluate the throughput of HiLS PA scheme with different input scale (n) and various subblock size (m). The result is shown in Fig. 4, where we set the input weak secure key length n equals from 16 Mb to 512 Mb, and splitting factor, defined as \(\frac{m}{n}\) is various from \(\frac{1}{32}\) to \(\frac{1}{2}\). Figure 4 shows us that for given n (in our implementation, can be up to 1 Gb), HiLS PA scheme can always achieve optimized throughput when splitting factor \(\frac{m}{n}=0.125\). When the splitting factor \(\frac{m}{n}\le 0.0625\), the Toeplitz matrix at least has to be divided into 28 submatrices with compression ratio \(\frac{r}{n}\ge 0.125\), larger than 16 (amount of total cores), resulting HiLS PA scheme with very poor throughput due to heavy overhead of complicated process scheduling. When the splitting factor \(\frac{m}{n}\ge 0.25\), less split submatrices (≤4) only contributes a bit speedup to HiLS PA scheme, due to not fully used computational resource and still large scaled FFT operations. When the splitting factor \(0.125 < \frac{m}{n} < 0.25\), the amount of split submatrices stays the level as the case with splitting factor equals to 0.125, but the FFT operating scale is same as the case with splitting factor equals to 0.25, which results even worse throughput to HiLS PA scheme. This situation would also happened when the splitting factor \(0.0625 < \frac{m}{n} < 0.125\). For example, when n = 512 Mb, the optimized throughput of HiLS PA scheme is 59.06 Mbps, 50.48 Mbps and 30.49 Mbps when the compression ratio equals to 0.125, 0.25 and 0.50 respectively.
According to the simulation results shown in Fig. 2, the maximum compression ratio required for PA schemes is (\(\frac{r}{n}\)) is 0.297 for 10 GHz entanglement based QKD systems. Then, we optimized the implementation of HiLS PA scheme with n = 1 Mb, 16 Mb, 128 Mb and 1 Gb with compression ratio equals to 0.125, 0.25 and 0.375 respectively and compared with other previous works designed for QKD systems, e.g. entanglement based systems, the results are shown in Fig. 5. S. Yang et al.^{27} and J. Constantin et al.^{28} both implemented PA schemes on FGPA platform by performing multiplication operations, achieved 64.0 Mbps and 41.0 Mbps throughput with compression ratio equals to 0.10. Q. Li et al. achieved the throughput of 116.0 Mbps with adaptive compression ratio by implementing FFT operation on FPGA platform^{30}. However, FPGA platform is not suitable for the implementation of PA schemes with ultralarge input scales (larger than the order of 10^{8}). B. Liu et al. achieved the throughput of 60 Mbps with input scale of 12.8 megabits by implementing the FFT enhanced PA scheme on MIC platform^{8}. Z. L. Yuan et al. achieved the throughput of 108.77 Mbps with the input scale supported up to 128 megabits by implementing the NTT based PA scheme on MIC platform^{31}. Z. L. Yuan et al. also evaluated the performance of their PA scheme on CPU platform, resulting in the throughput of 28.22 Mbps. When the input scale is 128 Mb, the finitesizeeffect for the final secure key can be almost perfectly avoided, and the throughput of our proposed HiLS PA scheme reaches up to 71.16 Mbps, 54.08 Mbps and 39.15 Mbps with the compression ratio equals to 0.125, 0.25 and 0.375 respectively. In the case of input scale is 1 Gb, the throughput of HiLS PA scheme reaches up to 32.49 Mbps and 15.0 Mbps with the compression ratio equals to 0.125 and 0.25, which contributes much rigorous statistical fluctuation analysis and is remarkable higher than the required throughput when the total channel loss is expected larger than 87.6 dB.
With limited resource (128 GB memory, 1 TB storage and 16 CPU cores in total), the HiLS PA scheme with input scale of 128 Gb and the compression ratio equals to 0.125, runs around 83 hours, resulting a throughput of 0.44 Mbps. The implementation of PA with such large inputs on GPU platform is very difficult due to the complicated computation and memory scheduling strategies. Meanwhile, the throughput of the HiLS PA scheme can be easily improved on highspeed multicore CPU platforms with much larger configured memory.
Conclusion
In this paper, we propose a fast Fourier transform (FFT) enhanced highspeed and largescale (HiLS) PA scheme on multicore CPU platform. The long input weak secure key is divided into many blocks and the random seed for constructing Toeplitz matrix is shuffled to multiple subsequences respectively, then PA procedures are parallel implemented for all subkey blocks with correlated subsequences, afterwards the outcomes are merged as the final secure key. When the input scale is 128 Mb, our proposed HiLS PA scheme reaches 71.16 Mbps, 54.08 Mbps and 39.15 Mbps with the compression ratio equals to 0.125, 0.25 and 0.375 respectively, resulting achievable secure key generation rates close to the asymptotic limit. HiLS PA scheme can be efficiently implemented on the commercial CPU platform without increasing dedicated computational devices and can be applied to 10 GHz QKD systems with even larger input scales. The evaluated throughput of HiLS PA scheme is around 32.49 Mbps with the compression ratio equals to 0.125 and the input scale of 1 Gb, which is ten times larger than the previous works for QKD systems.Furthermore, with the limited computational resources, the achieved throughput of HiLS PA scheme is 0.44 Mbps with the compression ratio equals to 0.125, when the input scale equals up to 128 Gb. As randomness extraction with Toeplitz hashing in QRNG is particularly efficient, the HiLS PA scheme can be also performed in highspeed QRNG.
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Acknowledgements
This work was supported in part by the National High Technology Research and Development Program of China under Grant No. 2015AA1138 and the National Natural Science Foundation of China under Grant No. 61972410.
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B.Y.T. and B.L. proposed the scheme, performed the experiments, wrote the paper and contributed equally. Y.P.Z. helped with the experimental implementation and results analysis. This work was conceived by B.L. and W.R.Y., supervised by W.R.Y. and cosupervised by C.Q.W. All authors reviewed the manuscript.
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Tang, BY., Liu, B., Zhai, YP. et al. Highspeed and Largescale Privacy Amplification Scheme for Quantum Key Distribution. Sci Rep 9, 15733 (2019). https://doi.org/10.1038/s41598019502901
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DOI: https://doi.org/10.1038/s41598019502901
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