﻿ 基于混沌加密的水声扩频通信技术
 舰船科学技术  2022, Vol. 44 Issue (5): 101-104    DOI: 10.3404/j.issn.1672-7649.2022.05.020 PDF

Research on underwater acoustic spread spectrum communication technology based on chaotic encryption
WEI Xiao-fei, HE Ru-long, CHEN Jin
College of Electronic Engineering, Naval University of Engineering, Wuhan 430033, China
Abstract: In order to improve the security of underwater information transmission, chaotic encryption technology is applied to underwater acoustic spread spectrum communication. In view of the limitation of general spread spectrum sequence in spread spectrum communication, the chaotic sequence of discrete chaotic map is introduced, and several common discrete chaotic maps are compared. It is found that Chebyshev chaotic map has obvious advantages over Tent map and Logistic map. The excellent characteristics of Chebyshev sequence with encryption property are evaluated by simulation, and its effectiveness is verified.
Key words: chaotic encryption     spread spectrum communication     Chebyshev map
0 引　言

1 混沌映射加密通信技术

 图 1 基于混沌同步的保密通信 Fig. 1 Secure communication based on chaotic synchronization

1）Tent映射

Tent映射属于分段线性混沌映射。在映射函数的迭代规则下，经过有限次数的计算后（迭代次数取决于计算机设备），Tent映射任何一个初值条件都将失去效果，需要重新寻找新的映射函数初值。

 ${x_{k + 1}} = \left\{ \begin{gathered} {x_k}/\beta ,{\text{ }}{{\text{x}}_k} \in (0,\beta ]，\hfill \\ (1 - {x_k})/(1 - \beta ),{\text{ }}{{\text{x}}_k} \in (\beta ,1] 。\end{gathered} \right.$ (1)

2）Logistic映射

Logistic混沌离散映射函数，在一个有效初值的迭代计算下就能产生混沌序列。然而Logistic映射的不足之处也非常明显。有资料表明Logistic映射是最简单的混沌映射，且被广泛应用于保密通信中，对其混沌序列的恢复重建方法已引起相当重视，加密效果相对较差；由于Logistic映射不均匀的不变分布，映射函数得到的混沌实值序列在某些情况下会产生失衡，进一步会影响因之得到的二进制序列的平衡性，不符合扩频通信中扩频码的要求；其次只有Logistic方程中 $\mu$ 取最大值4时，才产生[0,1]上的满射（高度混沌）[8]

 ${x_{k + 1}} = \mu {x_k}(1 - {x_k}) ，$ (2)

3）Chebyshev映射

Chebyshev映射方程对初值极为敏感，因而较容易获得大量不同的实值序列。只要满足初值 ${x_0} \in [ - 1,1]$ ${x_0} \notin \{ \left. x \right|x = \cos (\text{π} q/p),p,q \in Z\}$ 的条件，Chebyshev混沌离散映射函数就能产生无周期的实值序列，且高度混沌。

2 Chebyshev混沌序列性能评估

1）Chebyshev混沌序列的构造

k阶Chebyshev离散映射方程[8]为：

 ${x}_{n+1}={\rm{cos}}（k\cdot {\rm{arccos}} {x}_{n}）\text{，}{x}_{0}\in [-1,1]。$ (3)

$k \geqslant 2$ 时，该映射即为混沌映射。当初始值 ${x_0} \in \{ \left. x \right|x =$ ${\rm{cos}} ({\text{π}} q/p),p,q \in Z\}$ 时，将通过迭代产生周期性序列[8]。而对于其他初始值 ${x_0}$ ，Chebyshev离散映射将对应产生均值为0的非周期序列，取值为（−1,1）。所以通常说几乎对于所有的初始值，映射将产生非周期序列[7-8]

2）Chebyshev实值序列性能仿真

①无周期性

 图 2 离散映射实值序列 Fig. 2 Discrete mapping real valued sequence

②平衡性

③初值敏感性

 图 3 初值不同的2个离散实值序列 Fig. 3 Two discrete real valued sequences with different initial values

④Chebyshev实值序列的相关性分析

 图 4 Chebyshev实值序列的自相关性 Fig. 4 Autocorrelation of Chebyshev real valued sequences

 图 5 序列的互相关性 Fig. 5 Cross correlation of sequences

⑤Chebyshev序列相关性与其参数kN的关系

 图 6 k值对Chebyshev序列相关性的影响 Fig. 6 Influence of k value on Chebyshev sequence correlation

 图 7 N值对Chebyshev序列相关性的影响 Fig. 7 Influence of N value on Chebyshev sequence correlation

3 结　语

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