搜索

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

n维离散超混沌系统及其在音频加密中的应用

周双 尹彦力 王诗雨 张盈谦

引用本文:
Citation:

n维离散超混沌系统及其在音频加密中的应用

周双, 尹彦力, 王诗雨, 张盈谦
cstr: 32037.14.aps.73.20241028

An n-dimensional discrete hyperchaotic system and its application in audio encryption

Zhou Shuang, Yin Yan-Li, Wang Shi-Yu, Zhang Ying-Qian
cstr: 32037.14.aps.73.20241028
PDF
HTML
导出引用
  • 与现有其他混沌系统不同, 本文探索了一种具有简单结构的$n$维离散超混沌系统. 首先, 利用正弦函数和幂函数以及简单运算构建n维离散超混沌系统; 然后, 理论分析该系统可以设置正的Lyapunov指数; 接下来, 以六维混沌系统为例, 对其进行了相图、分岔图、Lyapunov指数、复杂性等动力学特征分析; 最后, 将新的混沌系统结合前后项异或运算和真随机数构建一次一密音频加密新方法. 通过实验仿真, 所提出的算法与现有音频加密算法进行比较, 该算法具有更高安全性, 能够抵御各种常见攻击.
    Discrete chaotic system, as a pseudo-random signal source, plays a very important role in realizing secure communication. However, many low-dimensional chaotic systems are prone to chaos degradation. Therefore, many scholars have studied the construction of high-dimensional chaotic systems. However, many existing algorithms for constructing high-dimensional chaotic systems have relatively high time complexity and relatively complex structures. To solve this problem, this paper explores an n-dimensional discrete hyperchaotic system with a simple structure. Firstly, the n-dimensional discrete hyperchaotic system is constructed by using sine function and power function and simple operations. Then, it is theoretically analyzed based on Jacobian matrix method that the system can have the positive Lyapunov exponents. Next, the algorithm time complexity, sample entropy, correlation dimension and other indexes are compared with those of the existing methods. The experimental results show that our system has a simple structure, high complexity and good algorithm time complexity. Therewith, a six-dimensional chaotic system is chosen as an example, and the phase diagram, bifurcation diagram, Lyapunov expnonents, complexity and other characteristics of the system are analyzed. The results show that the proposed system has good chaotic characteristics. Moreover, to show the application of the proposed system, we apply it to audio encryption. According to this system, we combine it with the XOR operation and true random numbers to explore a novel method of one-cipher audio encryption. Through experimental simulation, compared with some existing audio encryption algorithms, this algorithm can satisfy various statistical tests and resist various common attacks. It is also validated that the proposed system can be effectively applied to the field of audio encryption.
      通信作者: 周双, zhoushuang@cqnu.edu.cn
    • 基金项目: 重庆市自然科学基金(批准号: CSTB2023NSCQ-MSX0401)资助的课题.
      Corresponding author: Zhou Shuang, zhoushuang@cqnu.edu.cn
    • Funds: Project supported by the Natural Science Foundation Project of Chongqing, China (Grant No. CSTB2023NSCQ-MSX0401).
    [1]

    Herbadji D, Herbadji A, haddad I, Kahia H, Belmeguenai A, Derouiche N 2024 Integration 97 102192Google Scholar

    [2]

    Joshi A B, Gaffar A 2024 Soft Comput. 28 5523Google Scholar

    [3]

    Yu F, Yu Q L, Chen H F, Kong X X, Molbel A A M, Cai S, Du S C 2022 Fractal Fract. 6 370Google Scholar

    [4]

    Wu R, Gao S, Wang X Y, Liu S B, Li Q, Erkan U, Tang X L 2022 Chaos, Soliton Fractals 165 112770Google Scholar

    [5]

    Wu R, Gao S, Iu H H, Zhou S, Erkan U, Toktas A, Tang X 2024 IEEE Internet Things J. 11 10214Google Scholar

    [6]

    Rahul B, Kuppusamy K, Senthilrajan A 2023 Multimed. Tools Appl. 82 43729Google Scholar

    [7]

    Demirtaş M 2023 Orclever Proc. Res. Dev. 2 28Google Scholar

    [8]

    Liu H J 2023 Multimed. Tools Appl. 82 27973Google Scholar

    [9]

    Cao Y F, Liu H J 2024 Multimed. Tools Appl. 83 79377Google Scholar

    [10]

    Zhu P X, Yang Q G 2023 Proc. Am. Math. Soc. 151 5353Google Scholar

    [11]

    王兴元, 王明军 2007 物理学报 56 6843Google Scholar

    Wang X Y, Wang M J 2007 Acta Phys. Sin. 56 6843Google Scholar

    [12]

    Yang Q G, Zhu P X 2024 Int. J. Bifurcation Chaos 34 2450122Google Scholar

    [13]

    赵智鹏, 周双, 王兴元 2021 物理学报 70 230502Google Scholar

    Zhao Z P, Zhou S, Wang X Y 2021 Acta Phys. Sin. 70 230502Google Scholar

    [14]

    扶龙香, 贺少波, 王会海, 孙克辉 2022 物理学报 71 030501Google Scholar

    Fu L X, He S B, Wang H H, Sun K H 2022 Acta Phys. Sin. 71 030501Google Scholar

    [15]

    Hua Z Y, Zhang Y X, Bao H, Huang H J, Zhou Y C 2022 IEEE Trans. Circuits Syst. Regul. Pap. 69 784Google Scholar

    [16]

    Huang L L, Liu J, Xiang J H, Zhang Z F, Du X L 2022 Chaos, Soliton Fractals 160 112248Google Scholar

    [17]

    Fan C L, Ding Q 2022 Chaos, Soliton Fractals 161 112323Google Scholar

    [18]

    Zhao M D, Liu H J 2023 Int. J. Bifurcation Chaos 33 2350070Google Scholar

    [19]

    Zhang Y X, Hua Z Y, Bao H, Huang H J, Zhou Y C 2023 IEEE Trans. Syst. Man Cybern Syst. 53 6516Google Scholar

    [20]

    Ding D W, Zhu H F, Zhang H W, Yang Z L, Xie D 2024 Chaos, Soliton Fractals 185 115168Google Scholar

    [21]

    MacQueen J 1967 Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability, USA, June 21– July 18 1965 and December 27, 1965–January 7, 1966 p281

    [22]

    Wang C F, Fan C L, Ding Q 2018 Int. J. Bifurcation Chaos 28 1850084Google Scholar

    [23]

    Termonia Y 1984 Phys. Rev. A 29 1612Google Scholar

    [24]

    Richman J S, Moorman J R 2000 Am. J. Physiol-Heart C 278 H2039Google Scholar

    [25]

    Grassberger P, Procaccia I 1983 Phys. Rev. Lett. 50 346Google Scholar

    [26]

    Zeng J, Wang Y M, Li X Y, Guang Y R, Wang C F, Ding Q 2023 Phys. Scr. 98 075212Google Scholar

    [27]

    Liu R R, Liu H J, Zhao M D 2023 Integration 93 102071Google Scholar

    [28]

    Xing Y N, Zeng J, Dong W J, Zhang J, Guo P T, Ding Q 2024 Phys. Scr. 99 035231Google Scholar

    [29]

    Kumar A, Dua M 2023 Appl. Acoust. 203 109196Google Scholar

  • 图 1  (a) x1 -x2 -x3相图; (b) x4 -x5 -x6相图

    Fig. 1.  (a) x1 -x2 -x3 phase diagram; (b) x4 -x5 -x6 phase diagram.

    图 2  分岔图

    Fig. 2.  Bifurcation diagram.

    图 3  Lyapunov指数

    Fig. 3.  Lyapunov exponent.

    图 4  敏感性分析

    Fig. 4.  Sensitivity analysis

    图 5  音频加密与解密仿真(第1列为原始音频波形图; 第2列为加密音频波形图; 第3列为解密音频波形图) (a) 1-67152-A-17.wav的加密与解密; (b) 1-121951-A-8.wav的加密与解密; (c) 5-261464-A-23.wav的加密与解密

    Fig. 5.  Audio encryption and decryption simulation (the first column is the original audio waveform diagram; the second column is an encrypted audio waveform diagram; the third column decrypts the audio waveform diagram): (a) Encryption and decryption of 1-67152-A-17.wav; (b) encryption and decryption of 1-121951-A-8.wav; (c) encryption and decryption of 5-261464-A-23.wav.

    图 6  选择明文攻击 (a) 1-67152-A-17.wav XOR 1-121951-A-8.wav; (b) 5-261464-A-23.wav XOR 1-67152-A-17.wav; (c) 1-121951-A-8.wav XOR 5-261464-A-23.wav; (d) 加密后的 1-67152-A-17.wav XOR加密后的 1-121951-A-8.wav; (e) 加密后的 5-261464-A-23.wav XOR 加密后的1-67152-A-17.wav; (f) 加密后的 1-121951-A-8.wav XOR加密后的 5-261464-A-23.wav

    Fig. 6.  Chosen-plaintext attack: (a) 1-67152-A-17.wav XOR 1-121951-A-8.wav; (b) 5-261464-A-23.wav XOR 1-67152-A-17.wav; (c) 1-121951-A-8.wav XOR 5-261464-A-23.wav; (d) encrypted 1-67152-A-17.wav XOR encrypted 1-121951-A-8.wav; (e) Encrypted 5-261464-A-23.wav XOR Encrypted 1-67152-A-17.wav; (f) encrypted 1-121951-A-8.wav XOR encrypted 5-261464-A-23.wav.

    表 1  不同方法的时间复杂度对比

    Table 1.  Comparison of time complexity of different methods.

    离散混沌系统 元素加法个数 矩阵乘法复杂度 算法时间复杂度
    Wang et al., 2018[22] $n(n - 1)$ $O({n^3})$ $O({n^3})$
    Hua et al., 2022[15] ${{n(n - 1)}}/{2}$ $O({n^3})$ $O({n^3})$
    Fan et al., 2022[17] $n(n-1)$ $O({n^3})$ $O({n^3})$
    Liu et al., 2023[18] ${{n(n - 1)}}/{2}$ $O({n^3})$ $O({n^3})$
    本文方法 $ 3n-4 \; ({b_i} \ne 0)$
    $ 2 n - 4\; ({b_i}=0)$
    $O(n)$
    下载: 导出CSV

    表 2  不同维度下各个变量动力学指标平均值分析

    Table 2.  Analysis of average dynamic indicators of variables across different dimensions.

    维度KESECD
    30.43122.01202.1857
    40.42122.04462.8712
    50.43202.09732.2378
    60.42262.93212.7423
    下载: 导出CSV

    表 3  不同6D混沌系统的6个变量平均值对比

    Table 3.  Comparison of average values of six variables across different 6D chaotic systems.

    系统 CD SE
    Zhao et al., 2023[18] 2.8493 2.0191
    Zeng et al., 2023[26] 2.8115 2.2268
    Liu et al., 2023[27] 2.8893 2.6958
    Xing et al., 2024[28] 1.4833 1.6015
    新的混沌系统 2.7423 2.9321
    下载: 导出CSV

    表 4  TestU01测试结果(未通过的数量)

    Table 4.  TestU01 test results (number of failures).

    变量(增益1015) $ {x_1} $ $ {x_2} $ $ {x_3} $ $ {x_4} $ $ {x_5} $ $ {x_6} $
    SmallCrush(15) 0 0 0 0 2 0
    Crush(144) 6 4 7 6 78 4
    下载: 导出CSV

    表 5  原始音频相关系数

    Table 5.  Correlation coefficients of original audio files.

    音频 明文
    $ A_{n}, A_{n+1} $ $ A_{n}, A_{n+2} $ $ A_{n}, A_{n+3} $
    1-67152-A-17.wav 0.8980 0.6885 0.5066
    1-121951-A-8.wav 0.9695 0.8840 0.7583
    5-261464-A-23.wav 0.9559 0.8041 0.6811
    平均 0.9411 0.7922 0.6487
    下载: 导出CSV

    表 6  加密音频相关系数

    Table 6.  Correlation coefficients of encrypted audio files.

    音频 密文
    $ A_{n}, A_{n+1} $ $ A_{n}, A_{n+2} $ $ A_{n}, A_{n+3} $
    1-67152-A-17.wav –0.0029 –0.0036 0.0011
    1-121951-A-8.wav –0.0017 0.0017 –0.0013
    5-261464-A-23.wav 0.0020 0.0019 0.0027
    绝对值平均 0.0022 0.0024 0.0017
    下载: 导出CSV

    表 7  加密音频相关系数对比

    Table 7.  Comparison of correlation coefficients of encrypted audio files.

    算法 本文 Rahul et al., 2023[6] Kumar et al., 2023[29] Cao et al., 2024[9] Wu et al., 2024[5]
    相关系数 0.0022 0.000013 0.0019 0.002512 0.0013
    下载: 导出CSV

    表 8  加密音频的信息熵

    Table 8.  Information entropy of encrypted audio files

    音频 明文 密文
    1-67152-A-17.wav1.05777.9993
    1-121951-A-8.wav1.94807.9993
    5-261464-A-23.wav1.30217.9993
    平均1.43597.9993
    下载: 导出CSV

    表 9  加密效率

    Table 9.  Encryption efficiency.

    音频 大小 加密时间/s 速度/(s·KB–1)
    1-67152-A-17.wav4300.28090.00065
    1-121951-A-8.wav4300.20970.00048
    5-261464-A-23.wav4300.21670.00050
    平均4300.23570.00054
    下载: 导出CSV

    表 10  加密效率对比

    Table 10.  Comparison of encryption efficiency.

    算法 本文算法 Wu et al., 2024[5] Joshi et al., 2024[2] Kumar et al., 2023[29]
    速度/(s·KB–1) 0.00054 0.00030 1.8309 0.0041
    下载: 导出CSV
  • [1]

    Herbadji D, Herbadji A, haddad I, Kahia H, Belmeguenai A, Derouiche N 2024 Integration 97 102192Google Scholar

    [2]

    Joshi A B, Gaffar A 2024 Soft Comput. 28 5523Google Scholar

    [3]

    Yu F, Yu Q L, Chen H F, Kong X X, Molbel A A M, Cai S, Du S C 2022 Fractal Fract. 6 370Google Scholar

    [4]

    Wu R, Gao S, Wang X Y, Liu S B, Li Q, Erkan U, Tang X L 2022 Chaos, Soliton Fractals 165 112770Google Scholar

    [5]

    Wu R, Gao S, Iu H H, Zhou S, Erkan U, Toktas A, Tang X 2024 IEEE Internet Things J. 11 10214Google Scholar

    [6]

    Rahul B, Kuppusamy K, Senthilrajan A 2023 Multimed. Tools Appl. 82 43729Google Scholar

    [7]

    Demirtaş M 2023 Orclever Proc. Res. Dev. 2 28Google Scholar

    [8]

    Liu H J 2023 Multimed. Tools Appl. 82 27973Google Scholar

    [9]

    Cao Y F, Liu H J 2024 Multimed. Tools Appl. 83 79377Google Scholar

    [10]

    Zhu P X, Yang Q G 2023 Proc. Am. Math. Soc. 151 5353Google Scholar

    [11]

    王兴元, 王明军 2007 物理学报 56 6843Google Scholar

    Wang X Y, Wang M J 2007 Acta Phys. Sin. 56 6843Google Scholar

    [12]

    Yang Q G, Zhu P X 2024 Int. J. Bifurcation Chaos 34 2450122Google Scholar

    [13]

    赵智鹏, 周双, 王兴元 2021 物理学报 70 230502Google Scholar

    Zhao Z P, Zhou S, Wang X Y 2021 Acta Phys. Sin. 70 230502Google Scholar

    [14]

    扶龙香, 贺少波, 王会海, 孙克辉 2022 物理学报 71 030501Google Scholar

    Fu L X, He S B, Wang H H, Sun K H 2022 Acta Phys. Sin. 71 030501Google Scholar

    [15]

    Hua Z Y, Zhang Y X, Bao H, Huang H J, Zhou Y C 2022 IEEE Trans. Circuits Syst. Regul. Pap. 69 784Google Scholar

    [16]

    Huang L L, Liu J, Xiang J H, Zhang Z F, Du X L 2022 Chaos, Soliton Fractals 160 112248Google Scholar

    [17]

    Fan C L, Ding Q 2022 Chaos, Soliton Fractals 161 112323Google Scholar

    [18]

    Zhao M D, Liu H J 2023 Int. J. Bifurcation Chaos 33 2350070Google Scholar

    [19]

    Zhang Y X, Hua Z Y, Bao H, Huang H J, Zhou Y C 2023 IEEE Trans. Syst. Man Cybern Syst. 53 6516Google Scholar

    [20]

    Ding D W, Zhu H F, Zhang H W, Yang Z L, Xie D 2024 Chaos, Soliton Fractals 185 115168Google Scholar

    [21]

    MacQueen J 1967 Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability, USA, June 21– July 18 1965 and December 27, 1965–January 7, 1966 p281

    [22]

    Wang C F, Fan C L, Ding Q 2018 Int. J. Bifurcation Chaos 28 1850084Google Scholar

    [23]

    Termonia Y 1984 Phys. Rev. A 29 1612Google Scholar

    [24]

    Richman J S, Moorman J R 2000 Am. J. Physiol-Heart C 278 H2039Google Scholar

    [25]

    Grassberger P, Procaccia I 1983 Phys. Rev. Lett. 50 346Google Scholar

    [26]

    Zeng J, Wang Y M, Li X Y, Guang Y R, Wang C F, Ding Q 2023 Phys. Scr. 98 075212Google Scholar

    [27]

    Liu R R, Liu H J, Zhao M D 2023 Integration 93 102071Google Scholar

    [28]

    Xing Y N, Zeng J, Dong W J, Zhang J, Guo P T, Ding Q 2024 Phys. Scr. 99 035231Google Scholar

    [29]

    Kumar A, Dua M 2023 Appl. Acoust. 203 109196Google Scholar

  • [1] 马召召, 杨庆超, 周瑞平. 一种基于摄动理论的不连续系统Lyapunov指数算法. 物理学报, 2021, 70(24): 240501. doi: 10.7498/aps.70.20210492
    [2] 李清都, 郭建丽. 切换系统Lyapunov指数的算法及应用. 物理学报, 2014, 63(10): 100501. doi: 10.7498/aps.63.100501
    [3] 吴浩, 侯威, 王文祥, 颜鹏程. 试用Lyapunov指数探讨气候突变及其前兆信号. 物理学报, 2013, 62(12): 129204. doi: 10.7498/aps.62.129204
    [4] 李春来, 禹思敏. 一个新的超混沌系统及其自适应追踪控制. 物理学报, 2012, 61(4): 040504. doi: 10.7498/aps.61.040504
    [5] 臧鸿雁, 范修斌, 闵乐泉, 韩丹丹. S-盒的Lyapunov指数研究. 物理学报, 2012, 61(20): 200508. doi: 10.7498/aps.61.200508
    [6] 刘扬正, 林长圣, 李心朝, 刘海鹏, 王忠林. Logistic-Unified混杂混沌系统. 物理学报, 2011, 60(3): 030502. doi: 10.7498/aps.60.030502
    [7] 冯朝文, 蔡理, 康强, 彭卫东, 柏鹏, 王甲富. 基于单电子晶体管 - 金属氧化物场效应晶体管电路的离散混沌系统实现. 物理学报, 2011, 60(11): 110502. doi: 10.7498/aps.60.110502
    [8] 刘勇. 耦合系统的混沌相位同步. 物理学报, 2009, 58(2): 749-755. doi: 10.7498/aps.58.749
    [9] 贾红艳, 陈增强, 袁著祉. 一个大范围超混沌系统的生成和电路实现. 物理学报, 2009, 58(7): 4469-4476. doi: 10.7498/aps.58.4469
    [10] 唐良瑞, 李静, 樊冰. 一个新四维自治超混沌系统及其电路实现. 物理学报, 2009, 58(3): 1446-1455. doi: 10.7498/aps.58.1446
    [11] 刘扬正, 姜长生. 关联可切换超混沌系统的构建与特性分析. 物理学报, 2009, 58(2): 771-778. doi: 10.7498/aps.58.771
    [12] 刘明华, 冯久超. 一个新的超混沌系统. 物理学报, 2009, 58(7): 4457-4462. doi: 10.7498/aps.58.4457
    [13] 何四华, 杨绍清, 石爱国, 李天伟. 基于图像区域Lyapunov指数的海面舰船目标检测. 物理学报, 2009, 58(2): 794-801. doi: 10.7498/aps.58.794
    [14] 杨永锋, 吴亚锋, 任兴民, 秦卫阳, 支希哲, 裘焱. 基于最大Lyapunov指数预测的EMD端点延拓. 物理学报, 2009, 58(6): 3742-3746. doi: 10.7498/aps.58.3742
    [15] 于思瑶, 郭树旭, 郜峰利. 半导体激光器低频噪声的Lyapunov指数计算和混沌状态判定. 物理学报, 2009, 58(8): 5214-5217. doi: 10.7498/aps.58.5214
    [16] 张晓丹, 刘翔, 赵品栋. 一类延迟混沌系统沿主轴方向上Lyapunov指数的计算方法. 物理学报, 2009, 58(7): 4415-4420. doi: 10.7498/aps.58.4415
    [17] 张勇, 关伟. 基于最大Lyapunov指数的多变量混沌时间序列预测. 物理学报, 2009, 58(2): 756-763. doi: 10.7498/aps.58.756
    [18] 刘扬正, 姜长生, 林长圣, 孙 晗. 四维切换超混沌系统. 物理学报, 2007, 56(9): 5131-5135. doi: 10.7498/aps.56.5131
    [19] 王兴元, 王明军. 超混沌Lorenz系统. 物理学报, 2007, 56(9): 5136-5141. doi: 10.7498/aps.56.5136
    [20] 盛利元, 孙克辉, 李传兵. 基于切延迟的椭圆反射腔离散混沌系统及其性能研究. 物理学报, 2004, 53(9): 2871-2876. doi: 10.7498/aps.53.2871
计量
  • 文章访问数:  666
  • PDF下载量:  87
  • 被引次数: 0
出版历程
  • 收稿日期:  2024-07-24
  • 修回日期:  2024-08-26
  • 上网日期:  2024-09-19
  • 刊出日期:  2024-11-05

/

返回文章
返回