# Deep Learning-Based Security Verification for a Random Number Generator Using White Chaos

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Scheme

#### 2.1. White Chaos-Based NRNG Setup

_{1}and DFB

_{2}, respectively operating at bias currents of 15.6 mA and 15.3 mA, have threshold currents of 10.9 mA and 11.1 mA, respectively. The center wavelengths of DFB

_{1}and DFB

_{2}are 1549.73 nm and 1549.62 nm, respectively. The feedback strength is set to −8.1 dB for ECL

_{1}and −7.9 dB for ECL

_{2}. In addition, the feedback delays of both ECLs are 91.7 ns and 91.9 ns, respectively. With these parameters of the entropy source, the white chaos is generated by optical heterodyning. After quantization with the 8-bit ADC, a 320 Gb/s white chaos-based NRNG is realized by selecting 4 LSBs at 80-GHz sampling rate.

_{1}and the final output of the NRNG. These are done by applying a novel DL model to data extracted at both stages of the NRNG. Note that the security of ECL

_{1}is only evaluated due to the similarity of ECLs.

#### 2.2. DRNG Setup

^{24}, 2

^{26}, 2

^{28}, 2

^{30}). It is necessary to study the predictive capability of the DL model in discovering inherent and intricate dependencies.

#### 2.3. Data Collection and Preprocessing

#### 2.4. Deep Learning Model

^{N}, which is the number of all possible N-bit numbers in predicting the next value.

_{1}, h

_{2}, …, h

_{t-1}) ∈ ℝ

^{m}

^{×(t-1)}, a convolutional neural network (CNN) is used to improve the predictive performance of the model by employing CNN filters on the row vectors of H. The CNN has k filters, each of which has length of T. In addition, the CNN with a rectified linear unit activation function yield H

^{C}∈ ℝ

^{m}

^{×k}, where H

_{i}denotes the convolutional value of the i-th row vector of H. Then, the context vector is calculated as a weighted sum of row vectors of H

^{C}. The score function ƒ is defined to evaluate relevance between H

_{i}and h

_{t}:

_{t}is the present state of the LSTM output, and W

_{a}∈ ℝ

^{k}

^{×m}. The attention weight α

_{i}is realized by introducing a sigmoid activation function:

_{t}∈ ℝ

^{k}, the row vectors of H

^{C}is weighted by α

_{i}:

_{t}and h

_{t}to yield the output of the attention layer,

_{h}∈ ℝ

^{m}

^{×m}, W

_{v}∈ ℝ

^{m}

^{×k}.

#### 2.5. Model Training and Validation

#### 2.6. System Evaluation

_{pred}, against the baseline probability, P

_{b}, which is the highest probability of guessing a variable in the data. For a DL model, P

_{pred}is the probability of predicting the eleventh number correctly in the test set, according to the preceding ten consecutive numbers. That is, P

_{pred}is a percentage of all the correct predictions out of the total number of test predictions,

_{T}is the number of correct classifications, and N

_{F}is the number of incorrect classifications. The baseline probability P

_{b}is related to the minimum entropy of the distribution from which a random value is generated. In NIST Special Publication 800-90B [42], the min-entropy of an independent discrete variable X that takes values from a set A = (x

_{1}, x

_{2}, …, x

_{k}) with probability Pr(X = x

_{i}) = p

_{i}, for i = 1, …, k is described as:

_{b}, which is why it is considered to be the baseline probability. For instance, an N-bit random number from datasets extracted at a certain stage of the NRNG or LC-RNG has a uniform probability distribution, which means that the highest probability of guessing the output of RNGs is 1/2

^{N}. If the DL-based predictive model could give a higher prediction probability compared to the baseline probability, there exist hidden correlations in the data from the corresponding stage of RNGs. Contrarily, little is learned by the model, and the random numbers have strong resistance against the predictive DL model. On the other hand, compared with the statistical property tests, the performance of the predictive model is studied by learning deviations in the data with different level of complexity.

## 3. Experimental Results

^{24}is shown in Figure 5a.

^{24}, 2

^{26}, 2

^{28}, 2

^{30}, respectively. Please note that the seed for generating pseudo-random numbers in a training set is different from that in the corresponding test set. Evidently, the probability of correct prediction by the model, P

_{pred}, surpasses the baseline probability, P

_{b}, when the length of the training set exceeds the period of LC-RNG, i.e., M is less than 2

^{28}. In addition, the provided model still has P

_{pred}better than P

_{b}by more than 6 standard deviations, even if M is 2

^{28}, which is much larger than the length of the training set. Meanwhile, P

_{pred}decreases when M increases given the same size of training set. When M is 2

^{30}or larger, P

_{pred}is approximately equal to P

_{b}. It could be that the datasets with higher level of complexity make the model more difficult to detect the correlations among random numbers.

^{24}, 2

^{26}, respectively. However, the random numbers can pass 15 tests of the test suite when M is 2

^{28}, 2

^{30}, respectively. Compared with the corresponding results from Figure 6, the DL model still achieves a higher prediction probability than the baseline probability, when random numbers with the period of 2

^{28}can pass the NIST test suite successfully. Briefly, the DL-based predictive model has the advantage in detecting correlations among random numbers to some extent, compared to the results of the NIST test suite.

^{20}, 2

^{22}, 2

^{24}, 2

^{26}). The prediction results of these deep learning models on the LC-RNG with different periods are shown in Table 3. The baseline probability, P

_{b}, is still 0.39%, since 8-bit random numbers extracted from different periods follow the same uniform probability distribution. In Table 3, the simple RNN-based model has no advantage in detecting the intricate correlations among random numbers when M ≥ 2

^{22}. We speculate that the RNN-based model is subject to the problem of gradient disappearance during the training process, and has difficulty in discovering deterministic correlations. The FNN-based model and RCNN-based model can detect correlations in the data when M ≤ 2

^{26}, and give higher prediction accuracy than P

_{b}. However, the TPA-based model consistently achieves a prediction accuracy of more than 95% when M ≤ 2

^{24}, which is significantly better than the performance of other models. The model still detects the correlations, even though the length of the training set is less than M.

^{24}, which are shown in Table 4. The RNN-based model with simple configuration still shows the weak learning capability when L increases. The performance of FNN-based model and RCNN-based model becomes better as L increases. These results show that the longer the length of the training set is, the higher the prediction accuracy. In addition, the FNN-based model performs better than others when L = 3.2 × 10

^{6}, because it consumes most computational resources (trainable parameters) among these models. The TPA-based model gives an obvious advantage in learning the correlations when L increases, compared with the performance of others. Specifically, given the same length of the training data, the model achieves higher prediction accuracy than other models when L ≥ 6.4 × 10

^{6}. As shown above, the performance of the predictive model is investigated and demonstrated in this scenario.

_{1}(denoted as Data

_{1}) and the final output of the NRNG (denoted as Data

_{2}). The results of the prediction are also shown in Figure 6. For the ECL

_{1}stage, the predictive DL model achieves 9.54 ± 0.05% accuracy, which obviously surpasses P

_{b}in guessing the next random value. For the final output of the NRNG, P

_{pred}is extremely close to P

_{b}, i.e., the provided model learns no patterns in the training dataset. For both stages of the NRNG, the results given the DL model are consistent with these of the NIST test suite in Table 1. In other words, the predictive model does as well as the NIST test suite in this scenario.

_{1}are depicted in Figure 7, including the radio-frequency (RF) spectrum, and the autocorrelation function. The RF spectra of the chaos of the ECL

_{1}and the white chaos are depicted in Figure 7a1,b1, respectively. For the spectrum map of the chaotic ECL

_{1}, a dominant peak approximately at the relaxation frequency can be clearly observed, which is detrimental to the bandwidth and flatness of chaos of ECL

_{1}[17]. Furthermore, we can observe an obvious pattern of periodic modulation from the insert of Figure 7a1. Please note that the period equals the reciprocal of the feedback delay time. The periodic modulation is actually the time-delay signature (TDS) that destroys the unpredictability and randomness of entropy source. However, in Figure 7b1, the spectrum of the white chaos is flat and broadband, which is not subject to the dominant peak and the periodic modulation pattern. That is, the white chaos generated by optical heterodyning has the great potential in extracting high-speed and trusted random numbers.

_{1}and the white chaos, as depicted in Figure 7a2,b2, respectively. The autocorrelation trace of the chaotic ECL

_{1}shows an apparent correlation peak at the feedback delay in Figure 7a2. We speculate that the retention of four LSBs still preserves the TDS in raw data, which precludes its use as a random number generator. By comparison, after optical heterodyning, the correlation trace of the heterodyne signal has no correlation peak in Figure 7b2, which indicates the elimination of such time-delay signature by heterodyning of two chaotic ECLs. In addition, other methods [43,44,45] of eliminating the TDS also significantly improve the randomness of RNGs.

_{1}stage. Thus, we believe that TDS of the chaotic ECL

_{1}causes the correlations among the data, and then gives the predictive model more chances to learn any temporal information among the data. For the white chaos, TDS is eliminated by heterodyning of two chaotic ECLs, and no characteristics are shown in the frequency domain. Evidently, the model cannot learn any temporal pattern in the training dataset collected from the final output of the NRNG, i.e., P

_{pred}≈ P

_{b}. Therefore, the NRNG has the strong resistance against our predictive DL.

## 4. Discussion

^{28}. Note that random numbers with the period of LC-RNG of 2

^{28}can pass the NIST test suite successfully. Additionally, the model can also detect deterministic patterns caused by TDS in the ECL

_{1}stage of the NRNG. However, little is learned by the model when the period of LC-RNG is larger than 2

^{28}. The prediction ability of the model is limited by the basic DL architecture and its parameters, such as the length of the training set, the size of the input sequence, and so on. Apparently, the optimization of the parameters and the sophisticated and advanced neural networks can improve the prediction accuracy.

## 5. Conclusions

_{1}stage, the model learns deterministic correlations among the dataset, and achieves higher accuracy than the baseline probability in guessing the next random number. After optical heterodyning of both chaotic ECLs and minimal post-processing are introduced, the predictive model detects no patterns in the data; this is the first work showing that the NRNG has the strong resistance against DL. By analyzing the temporal properties of both stages, we find that TDS, causing the inherent correlations among the data, is the key to be learned and detected by DL. Finally, we conclude that DL-based predictive model is expected to provide an efficient supplement for evaluating the security and quality of RNGs.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

DL | Deep Learning |

NRNG | Non-deterministic Random Number Generator |

DRNG | Deterministic Random Number Generator |

TPA | Temporal Pattern Attention |

ECL | External-Cavity Semiconductor Laser |

RNG | Random Number Generator |

FNN | Feedforward Neural Network |

RNN | Recurrent Neural Network |

RCNN | Recurrent Convolutional Neural Network |

DFB | Distributed Feedback Semiconductor Laser |

PC | Polarization Controller |

FC | Fiber Coupler |

OI | Optical Isolator |

VA | Variable Attenuator |

BPD | Balanced Photo-Detector |

ADC | Analog-to-Digital Converter |

LSBs | Least Significant Bits |

LC-RNG | Linear Congruential Random Number Generator |

LSTM | Long Short-Term Memory |

GRU | Gated Recurrent Unit |

FC | Fully Connected |

CNN | Convolutional Neural Network |

MP | Max-Pooling |

RF | Radio-Frequency |

TDS | Time-Delay Signature |

DFT | Discrete Fourier Transform |

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**Figure 1.**Experimental scheme for evaluating the security of RNGs, which comprises data collection and preprocessing, model training and validation, and system evaluation.

**Figure 2.**The structure of a NRNG based on white chaos. DFB

_{1,2}: distributed feedback semiconductor laser; PC

_{1,2}: polarization controller; FC

_{1,2}, FC: fiber coupler; OI

_{1,2}: optical isolator; VA

_{1,2}: variable attenuator; M

_{1,2}: fiber mirror; BPD: balanced photo-detector; ADC: analog-to-digital converter; LSBs: least significant bits.

**Figure 3.**Data preprocessing in a conventional approach where 10 consecutive adjacent numbers within the random sequence are used as one input sequence, whereas the next number after the input sequence is used as the output (label). The new sequence and corresponding output are updated by shifting three positions in the dataset.

**Figure 4.**Deep learning model based on temporal pattern attention, which consists of a one-hot encoder, a LSTM layer, a temporal pattern attention (TPA) layer, and a fully connected (FC) layer.

**Figure 5.**Distribution of standardized numbers from RNGs with different stages. (

**a**–

**c**) represent the probability distribution of the data from the output of LC-RNG with the period of 2

^{24}, the output of the ECL

_{1}, and the output of the NRNG, respectively.

**Figure 6.**Prediction performance of the deep learning-based predictive model at different stages of LC-RNG and the white chaos-based NRNG.

**Figure 7.**Temporal properties of the chaos of the ECL

_{1}as well as the white chaos-based NRNG. (

**a1**,

**b1**) respectively represent the RF spectra of the chaos of the ECL

_{1}and the white chaos. (

**a2**,

**b2**) respectively represent the autocorrelation traces of the chaos of the ECL

_{1}and the white chaos.

**Table 1.**Results of NIST statistical test suite on the datasets at different stages of LC-RNG and the white chaos-based NRNG.

Statistical Tests | LC-RNG | NRNG | ||||
---|---|---|---|---|---|---|

M = 2^{24} | M = 2^{26} | M = 2^{28} | M = 2^{30} | Data_{1} | Data_{2} | |

Frequency | Success | Success | Success | Success | Failure | Success |

Block Frequency | Success | Success | Success | Success | Success | Success |

Cumulative Sums | Success | Success | Success | Success | Failure | Success |

Runs | Success | Success | Success | Success | Failure | Success |

Longest Run | Success | Success | Success | Success | Success | Success |

Rank | Success | Success | Success | Success | Success | Success |

FFT | Failure | Success | Success | Success | Success | Success |

Non-overlapping Template | Failure | Failure | Success | Success | Failure | Success |

Overlapping Template | Success | Success | Success | Success | Success | Success |

Universal | Success | Success | Success | Success | Success | Success |

Approximate Entropy | Failure | Success | Success | Success | Failure | Success |

Random Excursions | Success | Success | Success | Success | Success | Success |

Random Excursions Variant | Success | Success | Success | Success | Success | Success |

Serial | Failure | Success | Success | Success | Failure | Success |

Linear Complexity | Success | Success | Success | Success | Success | Success |

Total successful tests | 11/15 | 14/15 | 15/15 | 15/15 | 9/15 | 15/15 |

RNN-Based Model | FNN-Based Model | RCNN-Based Model |
---|---|---|

Input layer ^{1} | Input layer ^{1} | Input layer ^{1} |

RNN-256 + Tanh | FC-256 + Relu | CNN ^{2}-64 + Relu + MP-2 |

FC-256 + Softmax | FC-256 + Relu | CNN ^{3}-128 + Relu+ MP-2 |

/ | FC-256 + Softmax | LSTM-128 + Tanh |

/ | / | FC-256 + Softmax |

^{1}The input layer with a one-hot encoder;

^{2}the CNN with a filter length of 9;

^{3}the CNN with a filter length of 3.

Model | LC-RNG (Accuracy: %) | |||
---|---|---|---|---|

M = 2^{20} | M = 2^{22} | M = 2^{24} | M = 2^{26} | |

RNN-based model | 44.42 ± 0.02 | 0.39 ± 0.01 | 0.39 ± 0.01 | 0.39 ± 0.01 |

FNN-based model | 99.81 ± 0.01 | 88.62 ± 0.05 | 77.82 ± 0.05 | 1.88 ± 0.02 |

RCNN-based model | 86.91 ± 0.05 | 61.45 ± 0.07 | 18.79 ± 0.06 | 1.93 ± 0.01 |

TPA-based model | 99.86 ± 0.02 | 99.38 ± 0.06 | 95.53 ± 0.05 | 0.39 ± 0.01 |

Model | Length of Training Data (×10^{6}) | |||
---|---|---|---|---|

1.6 | 3.2 | 6.4 | 8.0 | |

RNN-based model | 0.39 ± 0.01 | 0.39 ± 0.01 | 0.39 ± 0.01 | 0.39 ± 0.01 |

FNN-based model | 0.39 ± 0.01 | 67.62 ± 0.05 | 74.65 ± 0.03 | 77.82 ± 0.05 |

RCNN-based model | 0.39 ± 0.01 | 0.39 ± 0.01 | 10.37 ± 0.03 | 18.79 ± 0.06 |

TPA-based model | 0.39 ± 0.01 | 1.03 ± 0.02 | 92.80 ± 0.03 | 95.53 ± 0.05 |

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## Share and Cite

**MDPI and ACS Style**

Li, C.; Zhang, J.; Sang, L.; Gong, L.; Wang, L.; Wang, A.; Wang, Y.
Deep Learning-Based Security Verification for a Random Number Generator Using White Chaos. *Entropy* **2020**, *22*, 1134.
https://doi.org/10.3390/e22101134

**AMA Style**

Li C, Zhang J, Sang L, Gong L, Wang L, Wang A, Wang Y.
Deep Learning-Based Security Verification for a Random Number Generator Using White Chaos. *Entropy*. 2020; 22(10):1134.
https://doi.org/10.3390/e22101134

**Chicago/Turabian Style**

Li, Cai, Jianguo Zhang, Luxiao Sang, Lishuang Gong, Longsheng Wang, Anbang Wang, and Yuncai Wang.
2020. "Deep Learning-Based Security Verification for a Random Number Generator Using White Chaos" *Entropy* 22, no. 10: 1134.
https://doi.org/10.3390/e22101134