# A Hardware Pseudo-Random Number Generator Using Stochastic Computing and Logistic Map

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## Abstract

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## 1. Introduction

## 2. Background

#### 2.1. Logistic Map

#### 2.2. Stochastic Computing

**Multiplication operation**. In SC, the multiplication of two stochastic bit streams can be realized by a simple gate circuit, which can save a lot of hardware resources in large-scale calculations. Unipolar multiplication only needs to pass the input bit streams through an AND gate to get the output. It should be noted that the two input bit streams must be guaranteed to be uncorrelated.

**Addition operation**. In SC, the unipolar bit stream represents values between [0, 1]. The numerical range of the result after the addition should be [0, 2], which is not within the indicated range, so the addition in the random calculation needs to be performed by a special operation, which is called as scaled addition. The scaled addition operation allows the multiplexer to scale the output to the normal range.

**Subtraction operation**. Its circuit structure is almost the same as an addition operation, where only a NOT gate is used between the second bitstream and the selector.

## 3. Enhanced Digital Logistic Map and Hardware Implementation

#### 3.1. Enhanced Digital Logistic Map

#### 3.2. Hardware Implementation of PRNG

## 4. Performance Analysis

#### 4.1. Performance Improvement of Digital Chaotic System

#### 4.2. Initial Value Sensitivity

#### 4.3. Chaotic Attractor

#### 4.4. Autocorrelation

#### 4.5. Approximate Entropy

#### 4.6. Histogram of Frequency Distribution

#### 4.7. NIST SP 800-22 Analyses

#### 4.8. TestU01 Test

#### 4.9. Area Overhead

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**The circuit structure of SCE. (

**a**) is a multiplier, (

**b**) is an adder, and (

**c**) is a subtractor.

**Figure 8.**Iterative values produced by the logistic map and the proposed method based on digital implementation. (

**a**) The original logistic map; (

**b**) The enhanced logistic map.

**Figure 9.**Sequences generated by enhanced logistic map using different initial values. (

**a**) Initial values of 0.25 and 0.250015; (

**b**) Initial values of 0 and 0.00012.

**Figure 10.**Comparison of chaotic attractors. (

**a**) The original logistic map; (

**b**) The logistic map of getting stuck in cycles; (

**c**) The enhanced logistic map.

**Figure 11.**Autocorrelation comparison based on different logistic maps. (

**a**) The original logistic map; (

**b**) The enhanced logistic map.

**Figure 12.**Histogram comparison based on different logistic maps. (

**a**) The original logistic map; (

**b**) The enhanced logistic map.

Chaotic System | Tent Map | Logistic Map | The Proposed System |
---|---|---|---|

Approximate entropy | 0.5170 | 0.6506 | 0.7112 |

Sub-Tests | p-Value | Result |
---|---|---|

Frequency | 0.213309 | Pass |

Universal | 0.554405 | Pass |

Serial | 0.739918 | Pass |

Rank | 0.441376 | Pass |

FFT | 0.902014 | Pass |

Runs | 0.350485 | Pass |

Longest Run | 0.911413 | Pass |

Block Frequency | 0.565945 | Pass |

Cumulative Sums | 0.534146 | Pass |

Overlapping Template | 0.395130 | Pass |

Non-overlapping Template | 0.066882 | Pass |

Approximate Entropy | 0.122325 | Pass |

Linear Complexity | 0.911413 | Pass |

Random Excursions | 0.652391 | Pass |

Random Excursions Variant | 0.381832 | Pass |

PRNGs | Rabbit | Alphabit |
---|---|---|

Original logistic map | 35/38 | 15/17 |

Enhanced logistic map | 38/38 | 17/17 |

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**MDPI and ACS Style**

Liu, J.; Liang, Z.; Luo, Y.; Cao, L.; Zhang, S.; Wang, Y.; Yang, S.
A Hardware Pseudo-Random Number Generator Using Stochastic Computing and Logistic Map. *Micromachines* **2021**, *12*, 31.
https://doi.org/10.3390/mi12010031

**AMA Style**

Liu J, Liang Z, Luo Y, Cao L, Zhang S, Wang Y, Yang S.
A Hardware Pseudo-Random Number Generator Using Stochastic Computing and Logistic Map. *Micromachines*. 2021; 12(1):31.
https://doi.org/10.3390/mi12010031

**Chicago/Turabian Style**

Liu, Junxiu, Zhewei Liang, Yuling Luo, Lvchen Cao, Shunsheng Zhang, Yanhu Wang, and Su Yang.
2021. "A Hardware Pseudo-Random Number Generator Using Stochastic Computing and Logistic Map" *Micromachines* 12, no. 1: 31.
https://doi.org/10.3390/mi12010031