离散Arnold变换改进及其在图像置乱加密中的应用

吴成茂

doi:10.7498/aps.63.090504

摘要

为了改善传统二维Arnold变换用于图像置乱加密的效果，提出了离散Arnold变换的改进方法，并将其用于图像置乱加密测试研究.该方法利用现有离散标准映射的构造思想，将传统离散二维Arnold变换表达式中第一个变换表达式所对应变换结果非线性融入第二个变换表达式，实现经典离散二维Arnold 变换的非线性去拟仿射化修改，以便快速改善图像置乱加密效果.数学证明改进方法不再保持现有离散二维Arnold 变换所具有的拟仿射不变性，但是改进变换仍是一种具有周期性的可逆映射，将其用于图像置乱加密时，利用其周期性或逆变换能恢复置乱前原图像.大量实验结果表明，本文所建议的改进方法是有效的，相比现有的离散Arnold变换更具有实用价值意义.

关键词:

Abstract

To improve the image scrambling and encryption effect in traditional two-dimensional discrete Arnold transform, a new nonlinear transform for image scrambling is proposed which improves the classical discrete Arnold transform with quasi-affine properties, and can be applied in image scrambling and encryption researching. This method first makes good use of the construction thought in classical discrete standard map, and embeds the nonlinear expressions of output results of one congruence equation for classical two-dimensional discrete Arnold transform into the input item of the other congruence equation for two-dimensional discrete Arnold transform. Then a new transform with good nonlinear characteristics is constructed on the basis of classical two-dimensional discrete Arnold transform in order to quickly improve the scrambling effect of the gray image. In the end, through mathematical proof it is shown that the proposed transform no longer has the quasi-affine invariance properties in the existing two-dimensional discrete Arnold transform, but it is still a reversible mapping with periodic properties; and when it is applied in image scrambling encryption, the original image can be restored from the scrambling and encryption in gray image for its periodic properties or inverse transform. Some experimental results show that the proposed nonlinear transform is effective, and can obtain better scrambling and encryption quality than the existing discrete two-dimensional Arnold transform, meanwhile it is more practical than the standard Arnold transform in view of security.

Keywords:

作者及机构信息

吴成茂

1.
西安邮电大学电子工程学院, 西安 710121

基金项目: 国家自然科学基金（批准号：90607008，61073106）和陕西省教育厅科研计划专项（批准号：2013JK1129）资助的课题.

Authors and contacts

Wu Cheng-Mao

1.
School of Electronic Engineering, Xi'an University of Posts and Telecommunications, Xi'an 710121, China

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 90607008, 61073106), and the Scientific Research Project of the Education Department of Shaanxi Province, China (Grant No. 2013JK1129).

参考文献

[1]	Arnold V I, Avez A 1968 Ergodic Problems in Classical Mechanics (New York: Benjamin) p286
[2]
[3]	Franks J 1977 Am. J. Math 99 1089
[4]	Dyson F J, Falk H 1992 Amer. Math Mon. 99 603
[5]
[6]	Behrends E, Fiedler B 1998 Ergod. theor. Dyn. Systems 18 331
[7]
[8]
[9]	Li P, Xu J W 2005 J. Cent. South Univ. Technol. 12 278
[10]	Chen F, Wong K W, Liao X F, Xiang T 2012 IEEE Trans. Inform. Theory 58 445
[11]
[12]
[13]	Bao J H, Yang Q G 2012 Nonlinear Dyn. 70 1365
[14]	Chen F, Wong K W, Liao X F, Xiang T 2013 IEEE Trans. Inform. Theory 59 3249
[15]
[16]
[17]	Kong T, Zhang D 2004 J. Software 15 1558 (in Chinese) [张涛, 张亶 2004 软件学报 15 1558]
[18]
[19]	Huang W B, Zhang D Dong G C 2008 Appl. Math. J. Chin. Univ. 23 99 (in Chinese) [黄外斌, 张亶, 董光昌 2008 高校应用数学学报 23 99]
[20]
[21]	Shao L P, Qin Z, Heng X C, Gao H J 2008 Acta Electron. Sin. 36 1355 (in Chinese) [邵利平, 覃征, 衡星辰, 高洪江 2008 电子学报 36 1355]
[22]	Zhou L M 2010 M.S. Dissertation (Ganzhou: Gannan Normal University) (in Chinese) [周利敏 2010 硕士论文 (赣州: 赣南师范学院)]
[23]
[24]
[25]	Pan C D, Pan C B 1998 Simple Number Theory (Beijing: Beijing University Press) p136 (in Chinese) [潘承洞, 潘承彪 1998 简明数论 (北京: 北京大学出版社) 第136页]
[26]
[27]	Qi D X 1999 J. North Chin. Uinv. Technol. 11 24 (in Chinese) [齐东旭 1999 北方工业大学学报 11 24]
[28]
[29]	Qi D X, Zou J C, Han X Y 2000 Sci. Chin. (Ser. E) 43 304
[30]	Chen G, Mao Y B, Chui C K 2004 Chaos, Soliton Fract. 21 749
[31]
[32]	Deng X, Zhao D 2011 Opt. Commun. 284 5623
[33]
[34]	Liu Z, Gong M, Dou Y, Liu E, Ashfag M, Dai J, Liu S 2011 Opt. Laser Engin. 50 246
[35]
[36]	Kanso K, Chebleh M 2012 Commun. Nonlinear Sci. Numer. Simul. 17 2943
[37]
[38]
[39]	Ye G D, Wong K W 2012 Nonlinear Dyn. 69 2079
[40]	Ma Z G, Qiu Y S 2003 J. Chin. Inst. Telecom. 24 51 (in Chinese) [马在光, 丘水生 2003 通信学报 24 51]
[41]
[42]	Zhao L, Liao X F, Xiang T, Xiao D 2010 Acta phys. Sin. 59 1507 (in Chinese) [赵亮, 廖晓峰, 向涛, 肖迪 2010 59 1507]
[43]
[44]	Yang L Z, Chen K F 2004 Sci. Chin. (Ser. F) 32 151
[45]
[46]	Li Y J, Ge J H, Li C L, Sun Z L 2010 J. Univ. Sci. Technol. 32 1630 (in Chinese) [李用江, 葛建华, 李昌利, 孙志林 2010 北京科技大学学报 32 1630]
[47]
[48]
[49]	Li Y J, Li C L, Ge J H, Sun Z L 2010 Comput. Sci. 37 278 (in Chinese) [李用江, 李昌利, 葛建华, 孙志林 2010 计算机科学 37 278]
[50]	Li Y J 2011 Ph. D. Dissertation (Xian: Xidian University) (in Chinese) [李用江 2011 博士论文 (西安:西安电子科技大学)]
[51]
[52]	Fransson J 2013 B. S. Dissertation (Smaland: Linnaeus University)
[53]
[54]	Wu Y, Agaian S, Noonan J P 2012 IEEE Sign. Process. Lett. (received)
[55]
[56]
[57]	Guo J S, Jin C H 2003 J. Chin. inst. telecom. 26 131 (in Chinese) [郭建胜, 金辰辉 2003 通信学报 26 131]
[58]
[59]	Liu T, Min L Q 2011 J. Wuhan Univ. (Nat. Sci. Ed.) 57 444 (in Chinese) [刘婷, 闵乐泉 2011 武汉大学学报(理科版) 57 444]
[60]	Zhang Q, Shen M F, Zhai Y K 2007 J. Data Acq. Process. 22 292 (in Chinese) [张琼, 沈民奋, 翟懿奎 2007 数据采集与处理 22 292]
[61]
[62]	Guan J, Ding Z Y, Duan X F 2013 J. Guilin Univ. Electron. Technol. 33 152 (in Chinese) [关健, 丁振亚, 段雪峰 2013 桂林电子科技大学学报 33 152]
[63]
[64]
[65]	Bao J H 2010 Ph. D. Dissertation (Guangzhou: South China University of technology) (in Chinese) [鲍江宏 2010 博士论文 (广州:华南理工大学)]
[66]
[67]	Rosen K H (translated by Xiao H G) 2009 Elementary Number Theory and Its Application (5th Ed.) (Beijing: China Machine Press) p133 (in Chinese) [罗申KH 著(夏洪刚译)2009 初等数论及其应用(第5版) (北京:机械工业出版社)第133页]
[68]
[69]	Gelfreich V 2000 Phys. D 136 266
[70]
[71]	Li C G, Han Z Z, Zhang H R 2003 Chin. J. Comput. 26 465 (in Chinese)[李昌刚, 韩正之, 张浩然 2003 计算机学报 26 465]
[72]
[73]	Chee S, Lee S, Park C, Sung S H 1999 Electron. Lett. 35 707
[74]	Shao L P, Qin Z, Gao H J, Heng X C 2007 Acta Electron. Sin. 35 1290 (in Chinese)[邵利平, 覃征, 高洪江, 衡星辰 2007 电子学报 35 1290]
[75]
[76]	Wu C K, Wang X M 1995 J. Xidian Univ. 22 94 (in Chinese) [武传坤, 王新梅 1995 西安电子科技大学学报 22 94]
[77]
[78]
[79]	Du Y Z, Ju Y, Wu W 2005 J. Hefei Univ. Technol. (Nat. Ed.) 28 592 (in Chinese) [杜奕智, 琚耀, 吴伟 2005 合肥工业大学学报(自然科学版) 28 592]
[80]	Jonathan K, Yehuda L (translated by Ren W) 2011 Introduction to modern cryptography: Principles and Protocols (Beijing: National Defense Industry Press) p138 (in Chinese) [乔纳森卡茨,耶胡达林德尔著(任伟译)2011 现代密码学原理与协议(北京:国防工业出版社) 第138页]
[81]
[82]
[83]	Peng F, Qiu S S, Long M 2005 J. South Chin. Univ. Technol. (Nat. Sci. Ed.) 33 20 (in Chinese) [彭飞, 丘水生, 龙敏 2005 华南理工大学学报 (自然科学版) 33 20]
[84]	Xu S J, Wang J Z 2008 Acta Phys. Sin. 57 37 (in Chinese)[徐淑奖, 王继志 2008 57 37]
[85]
[86]
[87]	Wang J, Jiang G P 2011 Acta Phys. Sin. 60 060503 (in Chinese)[王静, 蒋国平 2011 60 060503]
[88]	Sun F Y, Liu S T, L Z W 2007 Chin. Phys. 16 3616
[89]
[90]	Wang Z, Huang X, Li N, Song X N 2012 Chin. Phys. B 21 050506
[91]
[92]	Luo Y L, Du M H 2013 Chin. Phys. B 22 080503
[93]
[94]	Zhang L Y, Li C Q, Wong K K, Shu S, Chen G R 2012 J. Syst. Software 85 2077
[95]
[96]
[97]	Zhang Y W, Wang Y M, Shen Y B 2007 Sci. Chin.(Ser. F) 50 334

施引文献

[1]	Arnold V I, Avez A 1968 Ergodic Problems in Classical Mechanics (New York: Benjamin) p286
[2]
[3]	Franks J 1977 Am. J. Math 99 1089
[4]	Dyson F J, Falk H 1992 Amer. Math Mon. 99 603
[5]
[6]	Behrends E, Fiedler B 1998 Ergod. theor. Dyn. Systems 18 331
[7]
[8]
[9]	Li P, Xu J W 2005 J. Cent. South Univ. Technol. 12 278
[10]	Chen F, Wong K W, Liao X F, Xiang T 2012 IEEE Trans. Inform. Theory 58 445
[11]
[12]
[13]	Bao J H, Yang Q G 2012 Nonlinear Dyn. 70 1365
[14]	Chen F, Wong K W, Liao X F, Xiang T 2013 IEEE Trans. Inform. Theory 59 3249
[15]
[16]
[17]	Kong T, Zhang D 2004 J. Software 15 1558 (in Chinese) [张涛, 张亶 2004 软件学报 15 1558]
[18]
[19]	Huang W B, Zhang D Dong G C 2008 Appl. Math. J. Chin. Univ. 23 99 (in Chinese) [黄外斌, 张亶, 董光昌 2008 高校应用数学学报 23 99]
[20]
[21]	Shao L P, Qin Z, Heng X C, Gao H J 2008 Acta Electron. Sin. 36 1355 (in Chinese) [邵利平, 覃征, 衡星辰, 高洪江 2008 电子学报 36 1355]
[22]	Zhou L M 2010 M.S. Dissertation (Ganzhou: Gannan Normal University) (in Chinese) [周利敏 2010 硕士论文 (赣州: 赣南师范学院)]
[23]
[24]
[25]	Pan C D, Pan C B 1998 Simple Number Theory (Beijing: Beijing University Press) p136 (in Chinese) [潘承洞, 潘承彪 1998 简明数论 (北京: 北京大学出版社) 第136页]
[26]
[27]	Qi D X 1999 J. North Chin. Uinv. Technol. 11 24 (in Chinese) [齐东旭 1999 北方工业大学学报 11 24]
[28]
[29]	Qi D X, Zou J C, Han X Y 2000 Sci. Chin. (Ser. E) 43 304
[30]	Chen G, Mao Y B, Chui C K 2004 Chaos, Soliton Fract. 21 749
[31]
[32]	Deng X, Zhao D 2011 Opt. Commun. 284 5623
[33]
[34]	Liu Z, Gong M, Dou Y, Liu E, Ashfag M, Dai J, Liu S 2011 Opt. Laser Engin. 50 246
[35]
[36]	Kanso K, Chebleh M 2012 Commun. Nonlinear Sci. Numer. Simul. 17 2943
[37]
[38]
[39]	Ye G D, Wong K W 2012 Nonlinear Dyn. 69 2079
[40]	Ma Z G, Qiu Y S 2003 J. Chin. Inst. Telecom. 24 51 (in Chinese) [马在光, 丘水生 2003 通信学报 24 51]
[41]
[42]	Zhao L, Liao X F, Xiang T, Xiao D 2010 Acta phys. Sin. 59 1507 (in Chinese) [赵亮, 廖晓峰, 向涛, 肖迪 2010 59 1507]
[43]
[44]	Yang L Z, Chen K F 2004 Sci. Chin. (Ser. F) 32 151
[45]
[46]	Li Y J, Ge J H, Li C L, Sun Z L 2010 J. Univ. Sci. Technol. 32 1630 (in Chinese) [李用江, 葛建华, 李昌利, 孙志林 2010 北京科技大学学报 32 1630]
[47]
[48]
[49]	Li Y J, Li C L, Ge J H, Sun Z L 2010 Comput. Sci. 37 278 (in Chinese) [李用江, 李昌利, 葛建华, 孙志林 2010 计算机科学 37 278]
[50]	Li Y J 2011 Ph. D. Dissertation (Xian: Xidian University) (in Chinese) [李用江 2011 博士论文 (西安:西安电子科技大学)]
[51]
[52]	Fransson J 2013 B. S. Dissertation (Smaland: Linnaeus University)
[53]
[54]	Wu Y, Agaian S, Noonan J P 2012 IEEE Sign. Process. Lett. (received)
[55]
[56]
[57]	Guo J S, Jin C H 2003 J. Chin. inst. telecom. 26 131 (in Chinese) [郭建胜, 金辰辉 2003 通信学报 26 131]
[58]
[59]	Liu T, Min L Q 2011 J. Wuhan Univ. (Nat. Sci. Ed.) 57 444 (in Chinese) [刘婷, 闵乐泉 2011 武汉大学学报(理科版) 57 444]
[60]	Zhang Q, Shen M F, Zhai Y K 2007 J. Data Acq. Process. 22 292 (in Chinese) [张琼, 沈民奋, 翟懿奎 2007 数据采集与处理 22 292]
[61]
[62]	Guan J, Ding Z Y, Duan X F 2013 J. Guilin Univ. Electron. Technol. 33 152 (in Chinese) [关健, 丁振亚, 段雪峰 2013 桂林电子科技大学学报 33 152]
[63]
[64]
[65]	Bao J H 2010 Ph. D. Dissertation (Guangzhou: South China University of technology) (in Chinese) [鲍江宏 2010 博士论文 (广州:华南理工大学)]
[66]
[67]	Rosen K H (translated by Xiao H G) 2009 Elementary Number Theory and Its Application (5th Ed.) (Beijing: China Machine Press) p133 (in Chinese) [罗申KH 著(夏洪刚译)2009 初等数论及其应用(第5版) (北京:机械工业出版社)第133页]
[68]
[69]	Gelfreich V 2000 Phys. D 136 266
[70]
[71]	Li C G, Han Z Z, Zhang H R 2003 Chin. J. Comput. 26 465 (in Chinese)[李昌刚, 韩正之, 张浩然 2003 计算机学报 26 465]
[72]
[73]	Chee S, Lee S, Park C, Sung S H 1999 Electron. Lett. 35 707
[74]	Shao L P, Qin Z, Gao H J, Heng X C 2007 Acta Electron. Sin. 35 1290 (in Chinese)[邵利平, 覃征, 高洪江, 衡星辰 2007 电子学报 35 1290]
[75]
[76]	Wu C K, Wang X M 1995 J. Xidian Univ. 22 94 (in Chinese) [武传坤, 王新梅 1995 西安电子科技大学学报 22 94]
[77]
[78]
[79]	Du Y Z, Ju Y, Wu W 2005 J. Hefei Univ. Technol. (Nat. Ed.) 28 592 (in Chinese) [杜奕智, 琚耀, 吴伟 2005 合肥工业大学学报(自然科学版) 28 592]
[80]	Jonathan K, Yehuda L (translated by Ren W) 2011 Introduction to modern cryptography: Principles and Protocols (Beijing: National Defense Industry Press) p138 (in Chinese) [乔纳森卡茨,耶胡达林德尔著(任伟译)2011 现代密码学原理与协议(北京:国防工业出版社) 第138页]
[81]
[82]
[83]	Peng F, Qiu S S, Long M 2005 J. South Chin. Univ. Technol. (Nat. Sci. Ed.) 33 20 (in Chinese) [彭飞, 丘水生, 龙敏 2005 华南理工大学学报 (自然科学版) 33 20]
[84]	Xu S J, Wang J Z 2008 Acta Phys. Sin. 57 37 (in Chinese)[徐淑奖, 王继志 2008 57 37]
[85]
[86]
[87]	Wang J, Jiang G P 2011 Acta Phys. Sin. 60 060503 (in Chinese)[王静, 蒋国平 2011 60 060503]
[88]	Sun F Y, Liu S T, L Z W 2007 Chin. Phys. 16 3616
[89]
[90]	Wang Z, Huang X, Li N, Song X N 2012 Chin. Phys. B 21 050506
[91]
[92]	Luo Y L, Du M H 2013 Chin. Phys. B 22 080503
[93]
[94]	Zhang L Y, Li C Q, Wong K K, Shu S, Chen G R 2012 J. Syst. Software 85 2077
[95]
[96]
[97]	Zhang Y W, Wang Y M, Shen Y B 2007 Sci. Chin.(Ser. F) 50 334

[1]	郎利影, 陆佳磊, 于娜娜, 席思星, 王雪光, 张雷, 焦小雪. 基于深度学习的联合变换相关器光学图像加密系统去噪方法. , 2020, 69(24): 244204. doi: 10.7498/aps.69.20200805
[2]	姚丽莉, 袁操今, 强俊杰, 冯少彤, 聂守平. 基于gyrator变换和矢量分解的非对称图像加密方法. , 2016, 65(21): 214203. doi: 10.7498/aps.65.214203
[3]	刘啸天, 周国华, 李振华, 陈兴. 基于双缘调制的数字电压型控制Buck变换器离散迭代映射建模与动力学分析. , 2015, 64(22): 228401. doi: 10.7498/aps.64.228401
[4]	赵辽英, 吕步云, 厉小润, 陈淑涵. 基于尺度不变特征变换和区域互信息优化的多源遥感图像配准. , 2015, 64(12): 124204. doi: 10.7498/aps.64.124204
[5]	徐红梅, 金永镐, 金璟璇. 基于符号动力学的开关变换器时间不可逆性分析. , 2014, 63(13): 130502. doi: 10.7498/aps.63.130502
[6]	易奇志, 杜焰, 周天寿. 系统规模对群体行为的效果. , 2013, 62(11): 118701. doi: 10.7498/aps.62.118701
[7]	黎爱兵, 张立凤. 基于Lorenz映射的混沌系统分支变换预报规律研究. , 2013, 62(12): 120507. doi: 10.7498/aps.62.120507
[8]	马原, 吕群波, 刘扬阳, 钱路路, 裴琳琳. 基于主成分变换的图像稀疏度估计方法. , 2013, 62(20): 204202. doi: 10.7498/aps.62.204202
[9]	赵辽英, 马启良, 厉小润. 基于HIS 小波变换和MOPSO的全色与多光谱图像融合. , 2012, 61(19): 194204. doi: 10.7498/aps.61.194204
[10]	甘甜, 冯少彤, 聂守平, 朱竹青. 基于分块离散小波变换的图像信息隐藏与盲提取算法. , 2012, 61(8): 084203. doi: 10.7498/aps.61.084203
[11]	刘俊群. 斜置平面天线阵的近场-远场方向图分析. , 2012, 61(23): 238401. doi: 10.7498/aps.61.238401
[12]	甘甜, 冯少彤, 聂守平, 朱竹青. 基于分块DCT变换编码的小波域多幅图像融合算法. , 2011, 60(11): 114205. doi: 10.7498/aps.60.114205
[13]	曹光辉, 胡凯, 佟维. 基于Logistic均匀分布图像置乱方法. , 2011, 60(11): 110508. doi: 10.7498/aps.60.110508
[14]	赵亮, 廖晓峰, 向涛, 肖迪. 基于Z矩阵映射和选择加密的彩色图像退化算法研究. , 2010, 59(3): 1507-1523. doi: 10.7498/aps.59.1507
[15]	杨汝, 张波, 赵寿柏, 劳裕锦. 基于符号时间序列方法的开关变换器离散映射算法复杂度分析. , 2010, 59(6): 3756-3762. doi: 10.7498/aps.59.3756
[16]	刘奎, 丁宏林, 张贤高, 余林蔚, 黄信凡, 陈坤基. 量子点浮置栅量子线沟道三栅结构单电子场效应管存储特性的数值模拟. , 2008, 57(11): 7052-7056. doi: 10.7498/aps.57.7052
[17]	张晓明, 彭建华, 张入元. 利用线性可逆变换增强延迟反馈方法控制混沌的有效性. , 2005, 54(7): 3019-3026. doi: 10.7498/aps.54.3019
[18]	楼森岳. 利用Miura型不可逆变换得到高维可积模型. , 2000, 49(9): 1657-1662. doi: 10.7498/aps.49.1657
[19]	吴锋民, 朱启鹏, 施建青, 吴自勤. 有限步扩散反应置限分形聚集. , 1998, 47(4): 542-550. doi: 10.7498/aps.47.542
[20]	陈良恒. 不可逆化学反应扩散方程. , 1981, 30(7): 857-865. doi: 10.7498/aps.30.857

计量

文章访问数: 7406
PDF下载量: 775
被引次数: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

搜索

留言板