# A New Lightweight Stream Cipher Based on Chaos

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## Abstract

**:**

## 1. Introduction

## 2. Chaotic Sequence and Quantization

## 3. Logic Lightweight Stream Cipher

## 4. Design Principles

#### 4.1. Two NFSRs

#### 4.2. Digital Chaotic Module

#### 4.3. Filter Function

#### 4.4. Multiplexer Unit

## 5. Entropy Analyses

#### 5.1. Permutation Entropy

- There is a discrete time series $x(1)$, $x(2)$, …, $x(N)$ with length N, then an embedding dimension $m$ and a time delay τ are specified.
- By reconstructing the original sequence, each sub-sequence is represented as $X(i)$, and $X(i)=x(i)$, $x(i+\tau )$, …, $x(i+(m-1)\tau )$.
- Subsequently, incremental sorting is performed on each interior $X(i)$, i.e., $x(i+({j}_{1}-1)\tau )\le x(i+({j}_{2}-1)\tau )\le \dots \le x(i+({j}_{m}-1)\tau )$, if the two values are equal, the order is based on the subscripts $n$ in ${j}_{n}$. In this way, $X(i)$ is mapped to (j
_{1}, j_{2}, …, j_{m}), which is just one of $m!$ permutations. In other words, each subsequence $X(i)$ of dimension $m$ is mapped to one of $m!$ permutations. - Through the above steps, the continuous $m$ dimensional subspace is represented by a sequence of such symbols, in which the number of these symbols is $m!$. The probabilities of all symbols are expressed by ${p}_{1}$, ${p}_{2}$, …, ${p}_{k}$, where $k\le m!$.
- The permutation entropy of the time series $x(1)$, $x(2)$, …, $x(N)$ is:$$H(m)=-{\displaystyle \sum _{j=1}^{k}{p}_{j}\mathrm{ln}}{p}_{j}.$$

#### 5.2. Approximate Entropy

- Let $U(1)$, $U(2)$, …, $U(N)$ be a time series of dimension $N$, which is obtained by sampling at equal intervals.
- The relevant parameters $m$ and $r$ of the algorithm are defined, in which $m$ is an integer that represents the length of comparison vectors and $r$ is a real number using the measure of similarity.
- Here, the $m$ dimension vectors are reconstructed as $Y(1)$, $Y(2)$, …, $Y(N-m+1)$, where $Y(i)=[U(i),U(i+1),\dots ,U(i+m-1)]$.
- For $1\le i\le N-m+1$, the number of vectors satisfying the following conditions is counted.$${C}_{i}^{m}(r)=\frac{1}{N-m+1}SUM[d(i,j)\le r].$$

- 5.
- Let us define$${\mathsf{\Phi}}^{m}(r)=\frac{1}{N-m+1}{\displaystyle {\sum}_{i=1}^{N-m+1}\mathrm{log}}({C}_{i}^{m}(r)).$$
- 6.
- The approximate entropy (ApEn) is defined as$$ApEn={\mathsf{\Phi}}^{m}(r)-{\mathsf{\Phi}}^{m+1}(r).$$

#### 5.3. Information Entropy

## 6. Statistical Tests

## 7. Hardware Implementation Analysis

#### 7.1. Comparison of Implementation Results

#### 7.2. Throughtput Analysis

## 8. Security Evaluation

#### 8.1. Algebraic Attack

#### 8.2. TMDTO Attack

#### 8.3. Fault Attack

#### 8.4. Linear Approximation Attack

#### 8.5. Correlation Attack

## 9. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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Time Series | $\mathit{m}$ | $\mathit{\tau}$ | PE |
---|---|---|---|

Logistic | 3 | 1 | 0.3854 |

Logic | 3 | 1 | 0.5982 |

Time Series | $\mathit{m}$ | $\mathit{r}=0.2\mathit{s}\mathit{t}\mathit{d}$ | $\mathit{N}$ | ApEn |
---|---|---|---|---|

Logistic | 2 | 0.1013 | 2048 | 0.6655 |

Logic | 2 | 0.1015 | 2048 | 0.9442 |

Time Series | InEn |
---|---|

Logistic | 0.5951 |

Logic | 0.9238 |

Test | $\mathit{P}-\mathit{v}\mathit{a}\mathit{l}\mathit{u}\mathit{e}$ | Test |
---|---|---|

Frequency Test | 0.400908 | Success |

Frequency Test within a Block | 0.861626 | Success |

Runs Test | 0.475849 | Success |

Test for the Longest Run of Ones in a Block | 0.199175 | Success |

Binary Matrix Rank Test | 0.949536 | Success |

Discrete Fourier Transform Test | 0.232884 | Success |

Non-Overlapping Template Matching Test | 0.815009 | Success |

Overlapping Template Matching Test | 0.751585 | Success |

Maurer’s “Universal Statistical” Test | 0.139146 | Success |

Linear Complexity Test | 0.359316 | Success |

Serial Test | 0.067079 | Success |

Approximation Entropy Test | 0.011645 | Success |

Cumulative Sums Test | 0.557894 | Success |

Random Excursions Test | 0.459642 | Success |

Random Excursions Variant Test | 0.254816 | Success |

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**MDPI and ACS Style**

Ding, L.; Liu, C.; Zhang, Y.; Ding, Q.
A New Lightweight Stream Cipher Based on Chaos. *Symmetry* **2019**, *11*, 853.
https://doi.org/10.3390/sym11070853

**AMA Style**

Ding L, Liu C, Zhang Y, Ding Q.
A New Lightweight Stream Cipher Based on Chaos. *Symmetry*. 2019; 11(7):853.
https://doi.org/10.3390/sym11070853

**Chicago/Turabian Style**

Ding, Lina, Chunyuan Liu, Yanpeng Zhang, and Qun Ding.
2019. "A New Lightweight Stream Cipher Based on Chaos" *Symmetry* 11, no. 7: 853.
https://doi.org/10.3390/sym11070853