﻿ 结合多混沌映射与DNA的彩色图像加密算法
 计算机系统应用  2019, Vol. 28 Issue (12): 189-194 PDF

1. 水利部南京水利水文自动化研究所, 南京 210012;
2. 江苏南水科技有限公司, 南京 210012;
3. 云南省水文水资源局西双版纳分局, 西双版纳 666100

Wheel Selection Adaptive Image Encryption Algorithm Based on Multi-Chaotic Map and DNA
HU Chun-Jie1, HUANG Qi-Sheng3, CHEN Cui1, JI Hai-Xiang2, RUAN Cong2
1. Nanjing Automation Institute of Water Conservancy, Ministry of Water Resources, Nanjing 210012, China;
2. Jiangsu Nanshui Technology Co. Ltd., Nanjing 210012, China;
3. Xishuangbanna Branch, Yunnan Provincial Hydrographic and Water Resources Bureau, Xishuangbanna 666100, China
Abstract: Aiming at the problems of small space and low complexity of low-dimensional chaotic system and single DNA encryption scheme, a color image encryption algorithm based on multi-chaotic mapping and DNA is proposed. Firstly, Arnold transform is used to scramble the position of each component of the image. Logistic-sine chaotic map is used to generate random matrices of the same size as the plaintext image and block them. Then DNA rule operation is performed. The operation mode is determined dynamically by the chaotic sequence generated by Chen hyperchaotic system. The simulation results show that the algorithm has good encryption and recovery effect, can effectively resist various statistical attacks and differential attacks, and has good security, anti-noise, complex and high encryption performance.
Key words: Logistic mapping     position scrambling     DNA     hyper chaotic system

1 混沌系统 1.1 Logistic映射

Logistic映射是一个经典的非线性迭代方程, 其数学表达式如式(1)所示:

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

 图 1 系统状态随参数 $\mu$ 的演化图

1.2 Arnold映射

Arnold映射是一种非线性二维映射方程[9], 其公式定义如下:

 $\left[ {\begin{array}{*{20}{c}} {x'} \\ {y'} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&a \\ b&{ab + 1} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} x \\ y \end{array}} \right]od N$ (2)

1.3 Chen超混沌系统

Chen超混沌系统方程如下:

 $\left\{ {\begin{array}{*{20}{l}} {x = a(y - x) + w} \\ {y = dx - xz + cy} \\ {z = xy - bz} \\ {w = yz + ew} \end{array}} \right.$ (3)

 图 2 吸引子图

2 DNA编码技术

DNA中含4种不同的氮碱基分别是腺嘌呤A、胸腺嘧啶T、胞嘧啶C和鸟嘌呤G. 根据碱基互补配对原, 中A和T互补配对, C和G互补配对, 而数字图像中像素点的值可以用二进制表示, 在二进制中0和1是互补的, 因此00和11是互补的, 01和10是互补的. 基于这种思想, 结合二进制和DNA编码共有8种符合碱基编码规则, 如表1所示. 按照表1的方式, A用00表示, T用11表示, C用01来表示, G用10来表示. DNA的运算规则如表2~表4所示.

3 算法原理

 ${{g(k)}} = {{floor}}({{g(k)}} \times {10^3})\;{\rm{mod }}\;256$ (4)

 ${{x(k)}} = ({{floor}}(x(k) \times {10^4})od 8) + 1$ (5)
 ${{y(k)}} = ({{floor}}({{y}}(k) \times {10^4})od 8) + 1$ (6)

 ${{z(k)}} = {{floor}}({{z}}(k) \times {10^4})od 3$ (7)

z(k)=0时, 则图像矩阵与随机矩阵分块内所有像素一一对应进行DNA加法运算.

z(k)=1, 则图像矩阵与随机矩阵分块内所有像素一一对应进行减法运算.

z(k)=2为则图像矩阵与随机矩阵分块内所有像素一一对应进行异或运算.

4 仿真实验

5 算法分析 5.1 直方图分析

5.2 密钥空间分析

 图 3 图像加密结果

 图 4 图像加密前后的灰度值

5.3 信息熵

 $H(m) = \sum\limits_{i = 1}^{2N - 1} {P({m_i})} {\log _2}\frac{1}{{P({m_i})}}$ (8)

5.4 像素相关性分析

 \left\{ {\begin{aligned} & {D(x) = 1/n\sum\limits_{i = 1}^n {[{x_i} - E(x)]} } \\ & {\operatorname{cov} (x,y) = 1/n\sum\limits_{i = 1}^n {[{x_i} - E(x)][{y_i} - E(y)]} } \\ & {r = \operatorname{cov} (x,y)/(\sqrt {D(x)} \sqrt {D(y)} )} \end{aligned}} \right. (9)

5.5 抗噪声分析

 图 5 加入噪声后解密图像

6 结束语

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