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The concept of chaos cryptography or in the other words chaos-based cryptography can be divided into two major groups: the asymmetric<ref>{{Cite journal|date=2004-11-15|title=Public-key encryption with chaos|url=http://aip.scitation.org/doi/10.1063/1.1821671|journal=Chaos: An Interdisciplinary Journal of Nonlinear Science|volume=14|issue=4|pages=1078–1082|doi=10.1063/1.1821671|issn=1054-1500}}</ref><ref>{{Cite journal|last=Kocarev|first=L.|last2=Makraduli|first2=J.|last3=Amato|first3=P.|date=2005-10-01|title=Public-Key Encryption Based on Chebyshev Polynomials|url=https://link.springer.com/article/10.1007/s00034-005-2403-x|journal=Circuits, Systems and Signal Processing|language=en|volume=24|issue=5|pages=497–517|doi=10.1007/s00034-005-2403-x|issn=0278-081X}}</ref> and symmetric<ref name=":0">{{Cite journal|last=Akhavan|first=A.|last2=Samsudin|first2=A.|last3=Akhshani|first3=A.|date=2011-10-01|title=A symmetric image encryption scheme based on combination of nonlinear chaotic maps|url=http://www.sciencedirect.com/science/article/pii/S0016003211001104|journal=Journal of the Franklin Institute|volume=348|issue=8|pages=1797–1813|doi=10.1016/j.jfranklin.2011.05.001}}</ref><ref>{{Cite book|url=http://link.springer.com/chapter/10.1007/3-540-28247-5_8|title=Handbook of Geometric Computing|last=Mao|first=Yaobin|last2=Chen|first2=Guanrong|date=2005-01-01|publisher=Springer Berlin Heidelberg|isbn=9783540205951|pages=231–265|language=en|doi=10.1007/3-540-28247-5_8}}</ref><ref>{{Cite journal|last=Behnia|first=S.|last2=Akhshani|first2=A.|last3=Mahmodi|first3=H.|last4=Akhavan|first4=A.|date=2008-01-01|title=A novel algorithm for image encryption based on mixture of chaotic maps|url=http://www.sciencedirect.com/science/article/pii/S0960077906004681|journal=Chaos, Solitons & Fractals|volume=35|issue=2|pages=408–419|doi=10.1016/j.chaos.2006.05.011}}</ref> chaos-based cryptography. The majority of the symmetric chaos-based algorithms are based on the application of discrete chaotic maps in their process<ref>{{Cite journal|last=Behnia|first=Sohrab|last2=Akhshani|first2=Afshin|last3=Mahmodi|first3=Hadi|last4=Akhavan|first4=Amir|date=2008-01-01|title=Chaotic cryptographic scheme based on composition maps|url=http://www.worldscientific.com/doi/abs/10.1142/S0218127408020288|journal=International Journal of Bifurcation and Chaos|volume=18|issue=01|pages=251–261|doi=10.1142/S0218127408020288|issn=0218-1274}}</ref><ref name=":0" />.
The concept of chaos cryptography or in the other words chaos-based cryptography can be divided into two major groups: the asymmetric<ref>{{Cite journal|date=2004-11-15|title=Public-key encryption with chaos|url=http://aip.scitation.org/doi/10.1063/1.1821671|journal=Chaos: An Interdisciplinary Journal of Nonlinear Science|volume=14|issue=4|pages=1078–1082|doi=10.1063/1.1821671|issn=1054-1500}}</ref><ref>{{Cite journal|last=Kocarev|first=L.|last2=Makraduli|first2=J.|last3=Amato|first3=P.|date=2005-10-01|title=Public-Key Encryption Based on Chebyshev Polynomials|url=https://link.springer.com/article/10.1007/s00034-005-2403-x|journal=Circuits, Systems and Signal Processing|language=en|volume=24|issue=5|pages=497–517|doi=10.1007/s00034-005-2403-x|issn=0278-081X}}</ref> and symmetric<ref name=":0">{{Cite journal|last=Akhavan|first=A.|last2=Samsudin|first2=A.|last3=Akhshani|first3=A.|date=2011-10-01|title=A symmetric image encryption scheme based on combination of nonlinear chaotic maps|url=http://www.sciencedirect.com/science/article/pii/S0016003211001104|journal=Journal of the Franklin Institute|volume=348|issue=8|pages=1797–1813|doi=10.1016/j.jfranklin.2011.05.001}}</ref><ref>{{Cite book|url=http://link.springer.com/chapter/10.1007/3-540-28247-5_8|title=Handbook of Geometric Computing|last=Mao|first=Yaobin|last2=Chen|first2=Guanrong|date=2005-01-01|publisher=Springer Berlin Heidelberg|isbn=9783540205951|pages=231–265|language=en|doi=10.1007/3-540-28247-5_8}}</ref><ref>{{Cite journal|last=Behnia|first=S.|last2=Akhshani|first2=A.|last3=Mahmodi|first3=H.|last4=Akhavan|first4=A.|date=2008-01-01|title=A novel algorithm for image encryption based on mixture of chaotic maps|url=http://www.sciencedirect.com/science/article/pii/S0960077906004681|journal=Chaos, Solitons & Fractals|volume=35|issue=2|pages=408–419|doi=10.1016/j.chaos.2006.05.011}}</ref> chaos-based cryptography. The majority of the symmetric chaos-based algorithms are based on the application of discrete chaotic maps in their process<ref>{{Cite journal|last=Behnia|first=Sohrab|last2=Akhshani|first2=Afshin|last3=Mahmodi|first3=Hadi|last4=Akhavan|first4=Amir|date=2008-01-01|title=Chaotic cryptographic scheme based on composition maps|url=http://www.worldscientific.com/doi/abs/10.1142/S0218127408020288|journal=International Journal of Bifurcation and Chaos|volume=18|issue=01|pages=251–261|doi=10.1142/S0218127408020288|issn=0218-1274}}</ref><ref name=":0" />.


== Image Encryption ==
== Chaos-based Image Encryption ==
Bourbakis and Alexopoulos<ref>{{Cite journal|last=Bourbakis|first=N.|last2=Alexopoulos|first2=C.|title=Picture data encryption using scan patterns|url=http://linkinghub.elsevier.com/retrieve/pii/003132039290074S|journal=Pattern Recognition|language=en|volume=25|issue=6|pages=567–581|doi=10.1016/0031-3203(92)90074-s}}</ref> in 1991 proposed supposedly the earliest fully intended digital image encryption scheme which was based on SCAN language. Later on, with the emergence of chaos-based cryptography hundreds of new image encryption algorithms, all with the aim of improving the security of digital images were proposed<ref>{{Cite journal|last=Alvarez|first=Gonzalo|last2=Li|first2=Shujun|date=2006-08-01|title=Some basic cryptographic requirements for chaos-based cryptosystems|url=http://www.worldscientific.com/doi/abs/10.1142/S0218127406015970|journal=International Journal of Bifurcation and Chaos|volume=16|issue=08|pages=2129–2151|doi=10.1142/S0218127406015970|issn=0218-1274}}</ref>. However, there were three main aspects of the design of an image encryption that was usually modified in different algorithms (chaotic map, application of the map and structure of algorithm). The initial and perhaps most crucial point was the chaotic map applied in the design of the algorithms <ref>{{Cite journal|last=Behnia|first=S.|last2=Akhshani|first2=A.|last3=Ahadpour|first3=S.|last4=Mahmodi|first4=H.|last5=Akhavan|first5=A.|date=2007-07-02|title=A fast chaotic encryption scheme based on piecewise nonlinear chaotic maps|url=http://www.sciencedirect.com/science/article/pii/S0375960107002800|journal=Physics Letters A|volume=366|issue=4–5|pages=391–396|doi=10.1016/j.physleta.2007.01.081}}</ref><ref>{{Cite journal|last=Zhang|first=Leo Yu|last2=Hu|first2=Xiaobo|last3=Liu|first3=Yuansheng|last4=Wong|first4=Kwok-Wo|last5=Gan|first5=Jie|date=2014-10-01|title=A chaotic image encryption scheme owning temp-value feedback|url=http://www.sciencedirect.com/science/article/pii/S1007570414001415|journal=Communications in Nonlinear Science and Numerical Simulation|volume=19|issue=10|pages=3653–3659|doi=10.1016/j.cnsns.2014.03.016}}</ref><ref>{{Cite journal|last=Ghebleh|first=M.|last2=Kanso|first2=A.|date=2014-06-01|title=A robust chaotic algorithm for digital image steganography|url=http://www.sciencedirect.com/science/article/pii/S1007570413005030|journal=Communications in Nonlinear Science and Numerical Simulation|volume=19|issue=6|pages=1898–1907|doi=10.1016/j.cnsns.2013.10.014}}</ref><ref>{{Cite journal|last=Liu|first=Quan|last2=Li|first2=Pei-yue|last3=Zhang|first3=Ming-chao|last4=Sui|first4=Yong-xin|last5=Yang|first5=Huai-jiang|date=2015-02-01|title=A novel image encryption algorithm based on chaos maps with Markov properties|url=http://www.sciencedirect.com/science/article/pii/S1007570414002652|journal=Communications in Nonlinear Science and Numerical Simulation|volume=20|issue=2|pages=506–515|doi=10.1016/j.cnsns.2014.06.005}}</ref>. The speed of the cryptosystem is always an important parameter in the evaluation of the efficiency of a cryptography algorithm, therefore, the designers were initially interested in using simple chaotic maps such as tent map, and the logistic map. However, in 2006 and 2007, the new image encryption algorithms based on more suffisticated chaotic maps proved that application of chaotic map with higher dimension could improve the quality and security of the cryptosystems.
Bourbakis and Alexopoulos<ref>{{Cite journal|last=Bourbakis|first=N.|last2=Alexopoulos|first2=C.|title=Picture data encryption using scan patterns|url=http://linkinghub.elsevier.com/retrieve/pii/003132039290074S|journal=Pattern Recognition|language=en|volume=25|issue=6|pages=567–581|doi=10.1016/0031-3203(92)90074-s}}</ref> in 1991 proposed supposedly the earliest fully intended digital image encryption scheme which was based on SCAN language. Later on, with the emergence of chaos-based cryptography hundreds of new image encryption algorithms, all with the aim of improving the security of digital images were proposed<ref>{{Cite journal|last=Alvarez|first=Gonzalo|last2=Li|first2=Shujun|date=2006-08-01|title=Some basic cryptographic requirements for chaos-based cryptosystems|url=http://www.worldscientific.com/doi/abs/10.1142/S0218127406015970|journal=International Journal of Bifurcation and Chaos|volume=16|issue=08|pages=2129–2151|doi=10.1142/S0218127406015970|issn=0218-1274}}</ref>. However, there were three main aspects of the design of an image encryption that was usually modified in different algorithms (chaotic map, application of the map and structure of algorithm). The initial and perhaps most crucial point was the chaotic map applied in the design of the algorithms <ref>{{Cite journal|last=Behnia|first=S.|last2=Akhshani|first2=A.|last3=Ahadpour|first3=S.|last4=Mahmodi|first4=H.|last5=Akhavan|first5=A.|date=2007-07-02|title=A fast chaotic encryption scheme based on piecewise nonlinear chaotic maps|url=http://www.sciencedirect.com/science/article/pii/S0375960107002800|journal=Physics Letters A|volume=366|issue=4–5|pages=391–396|doi=10.1016/j.physleta.2007.01.081}}</ref><ref>{{Cite journal|last=Zhang|first=Leo Yu|last2=Hu|first2=Xiaobo|last3=Liu|first3=Yuansheng|last4=Wong|first4=Kwok-Wo|last5=Gan|first5=Jie|date=2014-10-01|title=A chaotic image encryption scheme owning temp-value feedback|url=http://www.sciencedirect.com/science/article/pii/S1007570414001415|journal=Communications in Nonlinear Science and Numerical Simulation|volume=19|issue=10|pages=3653–3659|doi=10.1016/j.cnsns.2014.03.016}}</ref><ref>{{Cite journal|last=Ghebleh|first=M.|last2=Kanso|first2=A.|date=2014-06-01|title=A robust chaotic algorithm for digital image steganography|url=http://www.sciencedirect.com/science/article/pii/S1007570413005030|journal=Communications in Nonlinear Science and Numerical Simulation|volume=19|issue=6|pages=1898–1907|doi=10.1016/j.cnsns.2013.10.014}}</ref><ref>{{Cite journal|last=Liu|first=Quan|last2=Li|first2=Pei-yue|last3=Zhang|first3=Ming-chao|last4=Sui|first4=Yong-xin|last5=Yang|first5=Huai-jiang|date=2015-02-01|title=A novel image encryption algorithm based on chaos maps with Markov properties|url=http://www.sciencedirect.com/science/article/pii/S1007570414002652|journal=Communications in Nonlinear Science and Numerical Simulation|volume=20|issue=2|pages=506–515|doi=10.1016/j.cnsns.2014.06.005}}</ref><ref name=":1">{{Cite journal|last=Behnia|first=S.|last2=Akhshani|first2=A.|last3=Akhavan|first3=A.|last4=Mahmodi|first4=H.|date=2009-04-15|title=Applications of tripled chaotic maps in cryptography|url=http://www.sciencedirect.com/science/article/pii/S0960077907006108|journal=Chaos, Solitons & Fractals|volume=40|issue=1|pages=505–519|doi=10.1016/j.chaos.2007.08.013}}</ref><ref>{{Cite journal|last=Kanso|first=A.|last2=Ghebleh|first2=M.|date=2015-07-01|title=An efficient and robust image encryption scheme for medical applications|url=http://www.sciencedirect.com/science/article/pii/S1007570414005668|journal=Communications in Nonlinear Science and Numerical Simulation|volume=24|issue=1–3|pages=98–116|doi=10.1016/j.cnsns.2014.12.005}}</ref>. The speed of the cryptosystem is always an important parameter in the evaluation of the efficiency of a cryptography algorithm, therefore, the designers were initially interested in using simple chaotic maps such as tent map, and the logistic map<ref>{{Cite journal|last=Kwok|first=H. S.|last2=Tang|first2=Wallace K. S.|date=2007-05-01|title=A fast image encryption system based on chaotic maps with finite precision representation|url=http://www.sciencedirect.com/science/article/pii/S0960077905011999|journal=Chaos, Solitons & Fractals|volume=32|issue=4|pages=1518–1529|doi=10.1016/j.chaos.2005.11.090}}</ref><ref>{{Cite journal|last=Baptista|first=M.S.|title=Cryptography with chaos|url=http://linkinghub.elsevier.com/retrieve/pii/S0375960198000863|journal=Physics Letters A|language=en|volume=240|issue=1-2|pages=50–54|doi=10.1016/s0375-9601(98)00086-3}}</ref>. However, in 2006 and 2007, the new image encryption algorithms based on more sophisticated chaotic maps proved that application of chaotic map with higher dimension could improve the quality and security of the cryptosystems <ref>{{Cite journal|last=Akhavan|first=Amir|last2=Mahmodi|first2=Hadi|last3=Akhshani|first3=Afshin|date=2006-11-01|title=A New Image Encryption Algorithm Based on One-Dimensional Polynomial Chaotic Maps|url=https://link.springer.com/chapter/10.1007/11902140_100|journal=Computer and Information Sciences – ISCIS 2006|language=en|publisher=Springer, Berlin, Heidelberg|pages=963–971|doi=10.1007/11902140_100}}</ref><ref>{{Cite journal|last=Akhshani|first=A.|last2=Mahmodi|first2=H.|last3=Akhavan|first3=A.|date=2006-10-01|title=A Novel Block Cipher Based on Hierarchy of One-Dimensional Composition Chaotic Maps|url=http://ieeexplore.ieee.org/document/4106949/|journal=2006 International Conference on Image Processing|pages=1993–1996|doi=10.1109/ICIP.2006.312889}}</ref><ref>{{Cite journal|last=Chuanmu|first=Li|last2=Lianxi|first2=H.|date=2007-04-01|title=A New Image Encryption Scheme based on Hyperchaotic Sequences|url=http://ieeexplore.ieee.org/document/4244820/|journal=2007 International Workshop on Anti-Counterfeiting, Security and Identification (ASID)|pages=237–240|doi=10.1109/IWASID.2007.373734}}</ref><ref name=":1" />.


==References ==
==References ==

Revision as of 19:40, 4 May 2017

Chaotic cryptography is the application of the mathematical chaos theory to the practice of the cryptography, the study or techniques used to privately and securely transmit information with the presence of a third-party or adversary. The use of chaos or randomness in cryptography has long been sought after by entities wanting a new way to encrypt messages. However, because of the lack of thorough, provable security properties and low acceptable performance, chaotic cryptography has encountered setbacks.

In order to use chaos theory acceptably in cryptography, they must first be mapped to each other. Properties in chaotic systems and cryptographic primitives share unique characteristics that allow for the chaotic systems to be applied to cryptography. If chaotic parameters, as well as cryptographic keys, can be mapped symmetrically or mapped to produce acceptable and functional outputs, it will make it next to impossible for an adversary to find the outputs without any knowledge of the initial values. Since chaotic maps in a real life scenario require a set of numbers that are limited, they may, in fact, have no real purpose in a cryptosystem if the chaotic behavior can be predicted. To counter this possibility, there exists simple to advanced ciphers. Chaos theory used in cryptosystems for commercial implementation has proven to be unsuccessful mainly because a chaos theories’ requirement to use intervals of real numbers. Given enough resources and time, an adversary could be able to predict functional outcomes. Since chaotic cryptosystems have no root in number theory this would make it difficult or impossible to implement therefore impractical.

Types of Chaos Cryptography

The concept of chaos cryptography or in the other words chaos-based cryptography can be divided into two major groups: the asymmetric[1][2] and symmetric[3][4][5] chaos-based cryptography. The majority of the symmetric chaos-based algorithms are based on the application of discrete chaotic maps in their process[6][3].

Chaos-based Image Encryption

Bourbakis and Alexopoulos[7] in 1991 proposed supposedly the earliest fully intended digital image encryption scheme which was based on SCAN language. Later on, with the emergence of chaos-based cryptography hundreds of new image encryption algorithms, all with the aim of improving the security of digital images were proposed[8]. However, there were three main aspects of the design of an image encryption that was usually modified in different algorithms (chaotic map, application of the map and structure of algorithm). The initial and perhaps most crucial point was the chaotic map applied in the design of the algorithms [9][10][11][12][13][14]. The speed of the cryptosystem is always an important parameter in the evaluation of the efficiency of a cryptography algorithm, therefore, the designers were initially interested in using simple chaotic maps such as tent map, and the logistic map[15][16]. However, in 2006 and 2007, the new image encryption algorithms based on more sophisticated chaotic maps proved that application of chaotic map with higher dimension could improve the quality and security of the cryptosystems [17][18][19][13].

References

  1. ^ "Public-key encryption with chaos". Chaos: An Interdisciplinary Journal of Nonlinear Science. 14 (4): 1078–1082. 2004-11-15. doi:10.1063/1.1821671. ISSN 1054-1500.
  2. ^ Kocarev, L.; Makraduli, J.; Amato, P. (2005-10-01). "Public-Key Encryption Based on Chebyshev Polynomials". Circuits, Systems and Signal Processing. 24 (5): 497–517. doi:10.1007/s00034-005-2403-x. ISSN 0278-081X.
  3. ^ a b Akhavan, A.; Samsudin, A.; Akhshani, A. (2011-10-01). "A symmetric image encryption scheme based on combination of nonlinear chaotic maps". Journal of the Franklin Institute. 348 (8): 1797–1813. doi:10.1016/j.jfranklin.2011.05.001.
  4. ^ Mao, Yaobin; Chen, Guanrong (2005-01-01). Handbook of Geometric Computing. Springer Berlin Heidelberg. pp. 231–265. doi:10.1007/3-540-28247-5_8. ISBN 9783540205951.
  5. ^ Behnia, S.; Akhshani, A.; Mahmodi, H.; Akhavan, A. (2008-01-01). "A novel algorithm for image encryption based on mixture of chaotic maps". Chaos, Solitons & Fractals. 35 (2): 408–419. doi:10.1016/j.chaos.2006.05.011.
  6. ^ Behnia, Sohrab; Akhshani, Afshin; Mahmodi, Hadi; Akhavan, Amir (2008-01-01). "Chaotic cryptographic scheme based on composition maps". International Journal of Bifurcation and Chaos. 18 (01): 251–261. doi:10.1142/S0218127408020288. ISSN 0218-1274.
  7. ^ Bourbakis, N.; Alexopoulos, C. "Picture data encryption using scan patterns". Pattern Recognition. 25 (6): 567–581. doi:10.1016/0031-3203(92)90074-s.
  8. ^ Alvarez, Gonzalo; Li, Shujun (2006-08-01). "Some basic cryptographic requirements for chaos-based cryptosystems". International Journal of Bifurcation and Chaos. 16 (08): 2129–2151. doi:10.1142/S0218127406015970. ISSN 0218-1274.
  9. ^ Behnia, S.; Akhshani, A.; Ahadpour, S.; Mahmodi, H.; Akhavan, A. (2007-07-02). "A fast chaotic encryption scheme based on piecewise nonlinear chaotic maps". Physics Letters A. 366 (4–5): 391–396. doi:10.1016/j.physleta.2007.01.081.
  10. ^ Zhang, Leo Yu; Hu, Xiaobo; Liu, Yuansheng; Wong, Kwok-Wo; Gan, Jie (2014-10-01). "A chaotic image encryption scheme owning temp-value feedback". Communications in Nonlinear Science and Numerical Simulation. 19 (10): 3653–3659. doi:10.1016/j.cnsns.2014.03.016.
  11. ^ Ghebleh, M.; Kanso, A. (2014-06-01). "A robust chaotic algorithm for digital image steganography". Communications in Nonlinear Science and Numerical Simulation. 19 (6): 1898–1907. doi:10.1016/j.cnsns.2013.10.014.
  12. ^ Liu, Quan; Li, Pei-yue; Zhang, Ming-chao; Sui, Yong-xin; Yang, Huai-jiang (2015-02-01). "A novel image encryption algorithm based on chaos maps with Markov properties". Communications in Nonlinear Science and Numerical Simulation. 20 (2): 506–515. doi:10.1016/j.cnsns.2014.06.005.
  13. ^ a b Behnia, S.; Akhshani, A.; Akhavan, A.; Mahmodi, H. (2009-04-15). "Applications of tripled chaotic maps in cryptography". Chaos, Solitons & Fractals. 40 (1): 505–519. doi:10.1016/j.chaos.2007.08.013.
  14. ^ Kanso, A.; Ghebleh, M. (2015-07-01). "An efficient and robust image encryption scheme for medical applications". Communications in Nonlinear Science and Numerical Simulation. 24 (1–3): 98–116. doi:10.1016/j.cnsns.2014.12.005.
  15. ^ Kwok, H. S.; Tang, Wallace K. S. (2007-05-01). "A fast image encryption system based on chaotic maps with finite precision representation". Chaos, Solitons & Fractals. 32 (4): 1518–1529. doi:10.1016/j.chaos.2005.11.090.
  16. ^ Baptista, M.S. "Cryptography with chaos". Physics Letters A. 240 (1–2): 50–54. doi:10.1016/s0375-9601(98)00086-3.
  17. ^ Akhavan, Amir; Mahmodi, Hadi; Akhshani, Afshin (2006-11-01). "A New Image Encryption Algorithm Based on One-Dimensional Polynomial Chaotic Maps". Computer and Information Sciences – ISCIS 2006. Springer, Berlin, Heidelberg: 963–971. doi:10.1007/11902140_100.
  18. ^ Akhshani, A.; Mahmodi, H.; Akhavan, A. (2006-10-01). "A Novel Block Cipher Based on Hierarchy of One-Dimensional Composition Chaotic Maps". 2006 International Conference on Image Processing: 1993–1996. doi:10.1109/ICIP.2006.312889.
  19. ^ Chuanmu, Li; Lianxi, H. (2007-04-01). "A New Image Encryption Scheme based on Hyperchaotic Sequences". 2007 International Workshop on Anti-Counterfeiting, Security and Identification (ASID): 237–240. doi:10.1109/IWASID.2007.373734.


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