一类混沌保密系统的设计及其实现
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摘要
本文把香农密码学的完全保密概念与混沌密码学联系起来,设计了一类理论上是完全保密的加密方案。在该保密方案中,采用了新的一类复合映射,证明了该类映射在Devaney定义下是混沌的,并讨论了该映射的不变分布和密码学特性。证明了该映射在有限分割生成的符号序列是独立均匀分布的,从而设计了一种加密结构使得加密方案理论上是完全保密方案。在实际应用中,由于有限精度计算,该类方案不可能真正达到完全保密。为了对抗有限精度效应,引入一类简单位操作使得加密方案在实际应用中也具有较强的抗破译能力。该类实际加密方案至少能有效对抗唯密文攻击、选择明文攻击。在对算法分析方面,至少能对抗符号动力学攻击、差分攻击。该算法具有一定的实用价值。本文还探讨了有限精度方面的问题,建立了有限精度下的该类方案的实现模型,对有限精度下的Lyapunov指数也做了简单讨论。最后,选取这类算法的一个具体算法应用于图像加密领域。基于图像的实验和性能分析指出该加密算法有较高的安全性。
This paper introduces the concept of the Shonnon’s perfect security into chaoticcryptography and designs a class of chaotic encryption systems who is perfectsecurity in theory. In these systems, we use a class of new compound chaoticmappings, which is chaotic by Deva ney chaotic theory. Then we discuss the invariantdistribution and the good cipher quality of these mappings. It’s proven that symbolicsequences generated using finite partition by these mappings have uniformdistribution and are independent of each other. The paper designs an encryptionarchitecture, which links with symbolic sequences are proven is perfect security intheory. In actual applica tion, to the limited precision computation, perfect security ofthe scheme is inaccessible . To fight the limited precision of computer, the paper addssome bit-ma nipulations in encryption algorithm process, this option can possessstrong anti-translated capability in actual applica tion. The actual encryption schemecan resist cipher text-only attack and known-pla intext attack effectively at least. Inalgorithm analysis , the scheme can resist symbolic dynamic analysis and differentialanalysis at least. Therefore, this class of the chaotic cryptographies have thepractica lly applied value . Besides, the article studies some theories about the limitedprecision of computer, establishes the limited precision model based on the schemeand discusses Lyapunov exponent in limited precision computer. At last, we apply anexample of a class of chaotic encryption systems to ima ge encryption . The statistica ltests and related analysis prove that the actual encryption scheme has high security.
This paper introduces the concept of the Shonnon’s perfect security into chaoticcryptography and designs a class of chaotic encryption systems who is perfectsecurity in theory. In these systems, we use a class of new compound chaoticmappings, which is chaotic by Deva ney chaotic theory. Then we discuss the invariantdistribution and the good cipher quality of these mappings. It’s proven that symbolicsequences generated using finite partition by these mappings have uniformdistribution and are independent of each other. The paper designs an encryptionarchitecture, which links with symbolic sequences are proven is perfect security intheory. In actual applica tion, to the limited precision computation, perfect security ofthe scheme is inaccessible . To fight the limited precision of computer, the paper addssome bit-ma nipulations in encryption algorithm process, this option can possessstrong anti-translated capability in actual applica tion. The actual encryption schemecan resist cipher text-only attack and known-pla intext attack effectively at least. Inalgorithm analysis , the scheme can resist symbolic dynamic analysis and differentialanalysis at least. Therefore, this class of the chaotic cryptographies have thepractica lly applied value . Besides, the article studies some theories about the limitedprecision of computer, establishes the limited precision model based on the schemeand discusses Lyapunov exponent in limited precision computer. At last, we apply anexample of a class of chaotic encryption systems to ima ge encryption . The statistica ltests and related analysis prove that the actual encryption scheme has high security.
引文
1王育明,刘建伟编著.通信网的安全—理论与技术.西安电子科技大学出版社, 1999: 140~178
2韦军.混沌序列密码的分析与设计.重庆大学博士学位论文. 2006: 20~100
3 Ljupco Kocarev. Chaos-based Cryptography: A Brief Verview. IEEE Circuits andSystems Maganize . 2001, 1(3): 6~21
4 A. M. Eskicioglu, J. Edward. An Overview of Multimed ia Content Protection inConsumer Electronic Device. Signal Processing: Ima ge Communica tion. 2001,(16): 681~699
5周红,周尚波.混沌理论在密码学中的应用.重庆大学学报.2004, 27(4): 39~436 M. G. Penedo, M. J. Carreira, A. Mosquera. Computer-aided Dia gnosis: ANeural-network Based Approach to Lung Nodule Detection. IEEE Transactions onMedica l Ima ging. 1998, 17(6): 872~880
7 X. G. Wu, H. P. Hu and B. L. Zha ng. Parameter Estima tion only from theSymbolic Sequences Generated by Chaos System. Chaos, Solitons and Fractals.2004, (22): 359~366
8佟晓筠.复合混沌序列密码在图像加密技术上的研究.哈尔滨工业大学博士论文. 2008: 10~93
9刘亚俊. http://www.sciencenet.cn/m/user_content.aspx?id=19786
10顾葆华.混沌系统的几种同步控制方法及其应用研究.南京理工大学博士论文. 2009: 30~56
11 O. E. . An Equation for Continuous Chaos. Physica l Letters A. 1976, Ro??ssler (57):397~398
12 L. O. Chua, M. Komuro, T. Matsumoto. The Double Scroll Family Part I:Rigorous Proof of Chaos. IEEE Transaction on Circuits&Systems I. 1986, (33): 1072~109613 G. R. Chen, T. Yet. Ueta. . Another Chaotic Attractor. Int. J. Bifurcation&Chaos.1999, (9): 1465~1466
14 J. H. , G. R. Chen, D. Z. Cheng. The Gap between the Loren z System and Lu?? theChen System. Int. J. Bifureation&Chaos. 2002, 12(12): 2917~2926
15 J. H. Lu?? , G. R. Chen, X. Yu. Dynamica l Beha viors of a Unified Chaotic System.Dynamica l of Continuous, Discrete and Impulsive Systems Series B. 2003, (1):115~124
16 C. X. Liu, T. Liu, L. Liu. A New Chaotic Attractor. Chaos, Solitons&Fracta ls.2004, (22): 1031~1038
17 C. X. Liu, T. Liu, L. Liu. A New Butterfly-shaped Attractor of Lorenz-like System.Chaos, Solitons&Fracta ls. 2006, (28): 1196~1203
18张宇辉,齐国元,刘文良.一个新的四维混沌系统理论分析与电路实现.物理学报. 2006, 55(7): 3307~3314
19 G. Y. Qi, S. Z. Du, G. R. Chen. On a Four-dimensiona l Chaotic System. Chaos,Solitons&Fracta ls. 2005, (23): 1671~1682
20 Y. X. Li, K. S. Tang, G. R. Chen. Generating Hyperchaos via State FeedbackControl. Int. J. Bifurcation&Chaos. 2005, 15(10): 3367~3375
21 G. Y. Qi, S. Z. Du, G. R. Chen. On a Four-dimensiona l Chaotic System. Chaos,Solitons&Fracta ls. 2005, (23): 1671~1682
22 A. M. Chen, J. A. Lu, J. H. Lu?? , S. M. Yu. Generating Hyperchaotic Lu?? Attractorvia State Feedback Control. Physics Letters A. 2006, (364): 103~110
23 T. G. Gao, G. R. Chen, Z. Q. Chen, S. J. Cang. The Generation and CircuitImplementation of a New Hyper-chaos Based upon Loren z System. PhysicsLetters A. 2007, (361): 78~ 86
24刘扬正,姜长生,林长圣.一类四维混沌系统切换混沌同步.物理学报. 2007,56(2): 707~712
25 H. J. Sun, H. J. Cao. Chaos Control and Synchronization of a Modified ChaoticSystem. Chaos, Solitons&Fracta ls. 2008, 37(5): 1442~1455
26 Y. Do, S. D. Kim, P. S. Kim. Stability of Fixed Points Placed on the Border in thePiece-wise Linear Systems . Chaos, Solitons&Fracta ls. 2008, 38(2): 391~399
27 I. Cervantes, J. C. Sanchez, F. J. Perez. Time and Resona nce Patterns in ChaoticPiece-wiselinear Systems . Chaos, Solitons&Fracta ls. 2008, 37(5): 1511~1527
28 F. Q. Wang, C. X. Liu. A New Multi-scroll Chaotic System. Chinese Physics.2006,15 (12): 2878~2882
29 W. Ahmad. A Simple Multi-scroll Hyperchaotic System. Chaos, Solitons&Fr-actal,2006, (7): 1213~1219
30 G. Y. Qi, M. A. Wyk, B. J. Wyk, et al. On a New Hperchaotic System. Physica lLetters A. 2008, 372(1): 124~136
31 L. M. Tam, J. H. Chen, H. K. Chen. Generation of Hyperchaos from the Chen-LeeSystem via Sinusoida l Perturbation. Chaos, Solitons&Fracta ls. 2008, 38(3): 526~83932 Z. M. Ge, C. H. Yang. Hyperchaos of Four State Autonomous System with ThreePositive Lyapunov Exponents. Physica l Letters A. 2008, (373): 349~353
33王光瑞,于熙龄,陈式刚.混沌控制、同步与利用.国防工业出版社, 2001: 23~100
34关新平,范正平,陈彩莲.混沌控制及其在保密通信中的应用.国防工业出版社, 2002: 1~75
35 J. M. Amigo, L. Kocarev, J. Szczepa nsk i. Theory and Practice of ChaoticCryptography. Physica l Letters A. 2007, (366): 211~216
36 R. Matthews . On the Derivation of a“Chaotic”Encryption Algorithm.Cryptologia. 1989, 13(1): 29~42
37 K. W. Wong, B. S. H. Kwok, W. S. Law. A Fast Ima ge Encryption Scheme Basedon Chaotic Standard Map. Physics Letters A. 2008, 372(15): 2645~2652
38 P. Li, Z. Li, A. H. Wolfga ng. Analysis of a Multiple-output Pseudo-Random-BitGenerator Based on a Spatiotemporal Chaotic System. Internationa l Journal ofBifurcation and Chaos. 2006, 16(10): 2949~2963
39 P. Li, Z. Li, A. H. Wolfga ng, et al. A Stream Cipher Based on a SpatiotemporalChaotic System. Chaos, Solitons and Fractals. 2007, (32): 1867~1876
40 L. H. Zha ng, Y. F. Zha ng. Research on Loren z Chaotic Stream Cipher. VLSIDesign and Video Technology 2005 Proceedings of 2005 IEEE Internationa lWorkshop: 431~434
41刘金梅,丘水生.混沌系统在密码学中的应用现状及展望.计算机工程与应用. 2008, 44(14): 5~12
42李昌刚,韩正之.图像加密技术新进展.信息与控制. 2003, 32(4): 339~343
43 X. Yi. Hash Function Based on Chaotic Tent Maps. IEEE Transactions on CircuitsSystems II. 2005, (52): 354~359
44 L. Kocarev, M. Sterjev, A. Fekete, G. Vattay. Public-key Encryption with Chaos .Chaos. 2004, 14(4): 1078~1082
45 N. Masuda, G. Jakimosk i, K. Aihara. Cryptosystems with Discretized ChaoticMaps. IEEE Transactions on Circuits Systems I. 2002, (53):1341~1350
46石峰,莫忠息.信息论基础.第二版.武汉大学出版社, 2006: 2~220
47 A. J. Menezes, P. C. Oorschot, S. A. Vanstone . Handbook ofApplied Cryp-tography. CRC Press LLC. 1997: 2~30
48 C. E. Shannon. Communica tion Theory of Secrecy Systems . Bell SystemTechnical Journal. July 1948, 28(4): 656~715
49 Whitfield Diffie , E. Martin. Hellma n. New Directions in Cryptography. IEEETransactions on Information Theory. 1976, 22(6): 644-654
50 R. L. Riverst, A. Shamir, L. Adlema n. A Method for Obtaining Digital Signaturesand Public Key Cryptosystem. Communica tions of the ACM. 1978, (21):120~126
51 T. ElGamal. A Public Key Crypto system and a Signature Scheme Based onDiscrete Logarithms . IEEE Transactions on Information Theory. 1985, (31): 469~47252 A. J. Menezes. Elliptic Curve Public Key Cryptosystems . Boston, Kluwer Acade -mic Publishers, 1993: 1~100
53 J. Banks, J. Braks, et al. On Deva ney’s Definition of Chaos. The America nMathema tical Monthly. 1992, 99(4): 332~334
54胡扬,阮炯.关于混沌Deva ney定义的一点注记.复旦大学学报. 1995, 34(2):127~131
55赵勇.关于两种混沌定义关系的探讨.汉中师范学院学报(自然科学学报).2003, 21(2): 4~6
56 Wen Huang, Xiangdong Ye. Deva ney’s Chaos or 2-scattering Implies Li–Yorke’sChaos. Topology and Its Applications . 2002, (117): 259–272
57 G. Alvarez, F. Montoya, M. Romera, et al. Crypta nalysis of an Ergodic ChaoticCipher. Physics Letters A. 1999, (263): 373~375
58 David Arroyo , Gonza lo Alvarez , Shujun Li . Some Hints for the Design of DigitalChaos-Based Cryptosystems: Lessons Learned from Crypta nalysis .arXiv:0812.0765v1
59 Gonza lo Alvarez, Shujun Li. Some Basic Cryptographic Requirements for Chaos-Based Cryptosystems . Internationa l Journal of Bifurcation and Chaos. 2006, 16(8):2129~2151
60 S. T. Fryska, M. A. Zohd y. Computer Dynamics and Shadowing of Chaotic Orbits.Physics Letters A. 1992, 166 (5 /6): 340~346
61 D. Maughan, M. Schertler, M. Schneider, et al. RFC 2408. Internet SecurityAssociation and Key Management Protocol (ISAKMP). 1998
62卢侃,孙建华,欧阳容百等.混沌动力学.上海翻译出版公司, 1990: 1~156
63徐辉,丁洪伟,赵一帆. 2007通信理论与技术新发展——第十二届全国青年通信学术会议论文集（下册）, 2007: 1182~1187
64凌聪,孙松庚.一种混沌扩频码序列设计的准则.电子科学学刊.1998, (4):558~561
65甘良才,吴燕翔.一类混沌映射产生调频序列的方法.电子学报. 2000, 28(4):109~111
66盛利元,肖燕予,盛喆.将混沌序列变换成均匀伪随机序列的普适算法.物理学报. 2008, 57( 7): 4007~4013
67 Xiaojun Tong, Minggen Cui. Ima ge Encryption with Compound ChaoticSequence Cipher Shifting Dynamica lly. Ima ge and Vision Computing. 2008, (26):843~850
68 G. Chen, Y. Mao, C. K. Chui. A Symmetric Ima ge Encryption Scheme Based On3D Chaotic Cat Maps. Chaos, Solitons and Fractals. 2004, 21(3): 749~761
69 A. Ukhin, Soto. Nechvatal, et al. SP 800-22. Statistica l Test Suite For Random andPseudora ndom Number Generators for Cryptographic Applications. 2001
2韦军.混沌序列密码的分析与设计.重庆大学博士学位论文. 2006: 20~100
3 Ljupco Kocarev. Chaos-based Cryptography: A Brief Verview. IEEE Circuits andSystems Maganize . 2001, 1(3): 6~21
4 A. M. Eskicioglu, J. Edward. An Overview of Multimed ia Content Protection inConsumer Electronic Device. Signal Processing: Ima ge Communica tion. 2001,(16): 681~699
5周红,周尚波.混沌理论在密码学中的应用.重庆大学学报.2004, 27(4): 39~436 M. G. Penedo, M. J. Carreira, A. Mosquera. Computer-aided Dia gnosis: ANeural-network Based Approach to Lung Nodule Detection. IEEE Transactions onMedica l Ima ging. 1998, 17(6): 872~880
7 X. G. Wu, H. P. Hu and B. L. Zha ng. Parameter Estima tion only from theSymbolic Sequences Generated by Chaos System. Chaos, Solitons and Fractals.2004, (22): 359~366
8佟晓筠.复合混沌序列密码在图像加密技术上的研究.哈尔滨工业大学博士论文. 2008: 10~93
9刘亚俊. http://www.sciencenet.cn/m/user_content.aspx?id=19786
10顾葆华.混沌系统的几种同步控制方法及其应用研究.南京理工大学博士论文. 2009: 30~56
11 O. E. . An Equation for Continuous Chaos. Physica l Letters A. 1976, Ro??ssler (57):397~398
12 L. O. Chua, M. Komuro, T. Matsumoto. The Double Scroll Family Part I:Rigorous Proof of Chaos. IEEE Transaction on Circuits&Systems I. 1986, (33): 1072~109613 G. R. Chen, T. Yet. Ueta. . Another Chaotic Attractor. Int. J. Bifurcation&Chaos.1999, (9): 1465~1466
14 J. H. , G. R. Chen, D. Z. Cheng. The Gap between the Loren z System and Lu?? theChen System. Int. J. Bifureation&Chaos. 2002, 12(12): 2917~2926
15 J. H. Lu?? , G. R. Chen, X. Yu. Dynamica l Beha viors of a Unified Chaotic System.Dynamica l of Continuous, Discrete and Impulsive Systems Series B. 2003, (1):115~124
16 C. X. Liu, T. Liu, L. Liu. A New Chaotic Attractor. Chaos, Solitons&Fracta ls.2004, (22): 1031~1038
17 C. X. Liu, T. Liu, L. Liu. A New Butterfly-shaped Attractor of Lorenz-like System.Chaos, Solitons&Fracta ls. 2006, (28): 1196~1203
18张宇辉,齐国元,刘文良.一个新的四维混沌系统理论分析与电路实现.物理学报. 2006, 55(7): 3307~3314
19 G. Y. Qi, S. Z. Du, G. R. Chen. On a Four-dimensiona l Chaotic System. Chaos,Solitons&Fracta ls. 2005, (23): 1671~1682
20 Y. X. Li, K. S. Tang, G. R. Chen. Generating Hyperchaos via State FeedbackControl. Int. J. Bifurcation&Chaos. 2005, 15(10): 3367~3375
21 G. Y. Qi, S. Z. Du, G. R. Chen. On a Four-dimensiona l Chaotic System. Chaos,Solitons&Fracta ls. 2005, (23): 1671~1682
22 A. M. Chen, J. A. Lu, J. H. Lu?? , S. M. Yu. Generating Hyperchaotic Lu?? Attractorvia State Feedback Control. Physics Letters A. 2006, (364): 103~110
23 T. G. Gao, G. R. Chen, Z. Q. Chen, S. J. Cang. The Generation and CircuitImplementation of a New Hyper-chaos Based upon Loren z System. PhysicsLetters A. 2007, (361): 78~ 86
24刘扬正,姜长生,林长圣.一类四维混沌系统切换混沌同步.物理学报. 2007,56(2): 707~712
25 H. J. Sun, H. J. Cao. Chaos Control and Synchronization of a Modified ChaoticSystem. Chaos, Solitons&Fracta ls. 2008, 37(5): 1442~1455
26 Y. Do, S. D. Kim, P. S. Kim. Stability of Fixed Points Placed on the Border in thePiece-wise Linear Systems . Chaos, Solitons&Fracta ls. 2008, 38(2): 391~399
27 I. Cervantes, J. C. Sanchez, F. J. Perez. Time and Resona nce Patterns in ChaoticPiece-wiselinear Systems . Chaos, Solitons&Fracta ls. 2008, 37(5): 1511~1527
28 F. Q. Wang, C. X. Liu. A New Multi-scroll Chaotic System. Chinese Physics.2006,15 (12): 2878~2882
29 W. Ahmad. A Simple Multi-scroll Hyperchaotic System. Chaos, Solitons&Fr-actal,2006, (7): 1213~1219
30 G. Y. Qi, M. A. Wyk, B. J. Wyk, et al. On a New Hperchaotic System. Physica lLetters A. 2008, 372(1): 124~136
31 L. M. Tam, J. H. Chen, H. K. Chen. Generation of Hyperchaos from the Chen-LeeSystem via Sinusoida l Perturbation. Chaos, Solitons&Fracta ls. 2008, 38(3): 526~83932 Z. M. Ge, C. H. Yang. Hyperchaos of Four State Autonomous System with ThreePositive Lyapunov Exponents. Physica l Letters A. 2008, (373): 349~353
33王光瑞,于熙龄,陈式刚.混沌控制、同步与利用.国防工业出版社, 2001: 23~100
34关新平,范正平,陈彩莲.混沌控制及其在保密通信中的应用.国防工业出版社, 2002: 1~75
35 J. M. Amigo, L. Kocarev, J. Szczepa nsk i. Theory and Practice of ChaoticCryptography. Physica l Letters A. 2007, (366): 211~216
36 R. Matthews . On the Derivation of a“Chaotic”Encryption Algorithm.Cryptologia. 1989, 13(1): 29~42
37 K. W. Wong, B. S. H. Kwok, W. S. Law. A Fast Ima ge Encryption Scheme Basedon Chaotic Standard Map. Physics Letters A. 2008, 372(15): 2645~2652
38 P. Li, Z. Li, A. H. Wolfga ng. Analysis of a Multiple-output Pseudo-Random-BitGenerator Based on a Spatiotemporal Chaotic System. Internationa l Journal ofBifurcation and Chaos. 2006, 16(10): 2949~2963
39 P. Li, Z. Li, A. H. Wolfga ng, et al. A Stream Cipher Based on a SpatiotemporalChaotic System. Chaos, Solitons and Fractals. 2007, (32): 1867~1876
40 L. H. Zha ng, Y. F. Zha ng. Research on Loren z Chaotic Stream Cipher. VLSIDesign and Video Technology 2005 Proceedings of 2005 IEEE Internationa lWorkshop: 431~434
41刘金梅,丘水生.混沌系统在密码学中的应用现状及展望.计算机工程与应用. 2008, 44(14): 5~12
42李昌刚,韩正之.图像加密技术新进展.信息与控制. 2003, 32(4): 339~343
43 X. Yi. Hash Function Based on Chaotic Tent Maps. IEEE Transactions on CircuitsSystems II. 2005, (52): 354~359
44 L. Kocarev, M. Sterjev, A. Fekete, G. Vattay. Public-key Encryption with Chaos .Chaos. 2004, 14(4): 1078~1082
45 N. Masuda, G. Jakimosk i, K. Aihara. Cryptosystems with Discretized ChaoticMaps. IEEE Transactions on Circuits Systems I. 2002, (53):1341~1350
46石峰,莫忠息.信息论基础.第二版.武汉大学出版社, 2006: 2~220
47 A. J. Menezes, P. C. Oorschot, S. A. Vanstone . Handbook ofApplied Cryp-tography. CRC Press LLC. 1997: 2~30
48 C. E. Shannon. Communica tion Theory of Secrecy Systems . Bell SystemTechnical Journal. July 1948, 28(4): 656~715
49 Whitfield Diffie , E. Martin. Hellma n. New Directions in Cryptography. IEEETransactions on Information Theory. 1976, 22(6): 644-654
50 R. L. Riverst, A. Shamir, L. Adlema n. A Method for Obtaining Digital Signaturesand Public Key Cryptosystem. Communica tions of the ACM. 1978, (21):120~126
51 T. ElGamal. A Public Key Crypto system and a Signature Scheme Based onDiscrete Logarithms . IEEE Transactions on Information Theory. 1985, (31): 469~47252 A. J. Menezes. Elliptic Curve Public Key Cryptosystems . Boston, Kluwer Acade -mic Publishers, 1993: 1~100
53 J. Banks, J. Braks, et al. On Deva ney’s Definition of Chaos. The America nMathema tical Monthly. 1992, 99(4): 332~334
54胡扬,阮炯.关于混沌Deva ney定义的一点注记.复旦大学学报. 1995, 34(2):127~131
55赵勇.关于两种混沌定义关系的探讨.汉中师范学院学报(自然科学学报).2003, 21(2): 4~6
56 Wen Huang, Xiangdong Ye. Deva ney’s Chaos or 2-scattering Implies Li–Yorke’sChaos. Topology and Its Applications . 2002, (117): 259–272
57 G. Alvarez, F. Montoya, M. Romera, et al. Crypta nalysis of an Ergodic ChaoticCipher. Physics Letters A. 1999, (263): 373~375
58 David Arroyo , Gonza lo Alvarez , Shujun Li . Some Hints for the Design of DigitalChaos-Based Cryptosystems: Lessons Learned from Crypta nalysis .arXiv:0812.0765v1
59 Gonza lo Alvarez, Shujun Li. Some Basic Cryptographic Requirements for Chaos-Based Cryptosystems . Internationa l Journal of Bifurcation and Chaos. 2006, 16(8):2129~2151
60 S. T. Fryska, M. A. Zohd y. Computer Dynamics and Shadowing of Chaotic Orbits.Physics Letters A. 1992, 166 (5 /6): 340~346
61 D. Maughan, M. Schertler, M. Schneider, et al. RFC 2408. Internet SecurityAssociation and Key Management Protocol (ISAKMP). 1998
62卢侃,孙建华,欧阳容百等.混沌动力学.上海翻译出版公司, 1990: 1~156
63徐辉,丁洪伟,赵一帆. 2007通信理论与技术新发展——第十二届全国青年通信学术会议论文集（下册）, 2007: 1182~1187
64凌聪,孙松庚.一种混沌扩频码序列设计的准则.电子科学学刊.1998, (4):558~561
65甘良才,吴燕翔.一类混沌映射产生调频序列的方法.电子学报. 2000, 28(4):109~111
66盛利元,肖燕予,盛喆.将混沌序列变换成均匀伪随机序列的普适算法.物理学报. 2008, 57( 7): 4007~4013
67 Xiaojun Tong, Minggen Cui. Ima ge Encryption with Compound ChaoticSequence Cipher Shifting Dynamica lly. Ima ge and Vision Computing. 2008, (26):843~850
68 G. Chen, Y. Mao, C. K. Chui. A Symmetric Ima ge Encryption Scheme Based On3D Chaotic Cat Maps. Chaos, Solitons and Fractals. 2004, 21(3): 749~761
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