Chaotic cryptology
{{Orphan|date=June 2017}}
Chaotic cryptology is the application of mathematical chaos theory to the practice of cryptography, the study or techniques used to privately and securely transmit information with the presence of a third-party or adversary. Since first being investigated by Robert Matthews in 1989,[https://www.tandfonline.com/doi/pdf/10.1080/0161-118991863745 "On the derivation of a “chaotic” encryption algorithm." Matthews, R.A.J. Cryptologia 13, no. 1 (1989): 29-42.] the use of chaos in cryptography has attracted much interest. However, long-standing concerns about its security and implementation speed continue to limit its implementation.[https://www.tandfonline.com/doi/abs/10.1080/0161-119191865821 "Supercomputer investigations of a chaotic encryption algorithm" DD Wheeler, RAJ Matthews Cryptologia 15 (2), 140-152]{{Cite journal|last1=Chen|first1=Yong|last2=Liao|first2=Xiaofeng|date=2005-07-25|title=Cryptanalysis on a modified Baptista-type cryptosystem with chaotic masking algorithm|journal=Physics Letters A|volume=342|issue=5–6|pages=389–396|doi=10.1016/j.physleta.2005.05.048|bibcode=2005PhLA..342..389C}}{{Cite journal|last1=Xie|first1=Eric Yong|last2=Li|first2=Chengqing|last3=Yu|first3=Simin|last4=Lü|first4=Jinhu|date=2017-03-01|title=On the cryptanalysis of Fridrich's chaotic image encryption scheme|journal=Signal Processing|volume=132|pages=150–154|doi=10.1016/j.sigpro.2016.10.002|arxiv=1609.05352|bibcode=2017SigPr.132..150X |s2cid=12416264}}{{Cite journal|last1=Akhavan|first1=A.|last2=Samsudin|first2=A.|last3=Akhshani|first3=A.|date=2015-09-01|title=Cryptanalysis of "an improvement over an image encryption method based on total shuffling"|journal=Optics Communications|volume=350|pages=77–82|doi=10.1016/j.optcom.2015.03.079|bibcode=2015OptCo.350...77A}}{{Cite journal|last1=Akhavan|first1=A.|last2=Samsudin|first2=A.|last3=Akhshani|first3=A.|date=2017-10-01|title=Cryptanalysis of an image encryption algorithm based on DNA encoding|journal=Optics & Laser Technology|volume=95|pages=94–99|doi=10.1016/j.optlastec.2017.04.022|bibcode=2017OptLT..95...94A}}
Chaotic cryptology consists of two opposite processes: Chaotic cryptography and Chaotic cryptanalysis. Cryptography refers to encrypting information for secure transmission, whereas cryptanalysis refers to decrypting and deciphering encoded encrypted messages.
In order to use chaos theory efficiently in cryptography, the chaotic maps are implemented such that the entropy generated by the map can produce required Confusion and diffusion. Properties in chaotic systems and cryptographic primitives share unique characteristics that allow for the chaotic systems to be applied to cryptography.{{Cite journal|last=Baptista|first=M.S.|title=Cryptography with chaos|journal=Physics Letters A|language=en|volume=240|issue=1–2|pages=50–54|doi=10.1016/s0375-9601(98)00086-3|year=1998|bibcode=1998PhLA..240...50B}} 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.{{Citation needed|date=March 2023}} 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.
One of the most important issues for any cryptographic primitive is the security of the system. However, in numerous cases, chaos-based cryptography algorithms are proved insecure.{{Cite book|last1=Li|first1=Shujun|last2=Zheng|first2=Xuan|title=2002 IEEE International Symposium on Circuits and Systems. Proceedings (Cat. No.02CH37353) |chapter=Cryptanalysis of a chaotic image encryption method |date=2002-01-01|volume=2|pages=II–708–II–711 vol.2|doi=10.1109/ISCAS.2002.1011451|isbn=978-0-7803-7448-5|s2cid=14523625|url=http://epubs.surrey.ac.uk/532413/1/ISCAS2002.pdf}}{{Cite journal|last1=Solak|first1=Ercan|last2=Çokal|first2=Cahit|last3=Yildiz|first3=Olcay Taner|last4=Biyikoğlu|first4=Türker|date=2010-05-01|title=Cryptanalysis of fridrich's chaotic image encryption|journal=International Journal of Bifurcation and Chaos|volume=20|issue=5|pages=1405–1413|doi=10.1142/S0218127410026563|issn=0218-1274|bibcode=2010IJBC...20.1405S|citeseerx=10.1.1.226.413}} The main issue in many of the cryptanalyzed algorithms is the inadequacy of the chaotic maps implemented in the system.{{cite arXiv|last1=Arroyo|first1=David|last2=Alvarez|first2=Gonzalo|last3=Fernandez|first3=Veronica|date=2008-05-28|title=On the inadequacy of the logistic map for cryptographic applications|eprint=0805.4355|class=nlin.CD}}{{Cite journal|last=Li|first=C.|title=Cracking a hierarchical chaotic image encryption algorithm based on permutation|journal=Signal Processing|volume=118|pages=203–210|doi=10.1016/j.sigpro.2015.07.008|arxiv=1505.00335|date=January 2016|bibcode=2016SigPr.118..203L |s2cid=7713295}}
Types
Chaos-based cryptography has been divided{{cite book |last1=Kocarev |first1=Ljupco |last2=Lian |first2=Shiguo |title=Chaos-Based Cryptography |date=2011 |publisher=Springer-Verlag |doi=10.1007/978-3-642-20542-2 |isbn=978-3-642-20542-2 |url=https://doi.org/10.1007/978-3-642-20542-2 |access-date=29 October 2021}} into two major groups:
- Symmetric chaos cryptography, where the same secret key is used by sender and receiver.{{Cite journal|last1=Akhavan|first1=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|journal=Journal of the Franklin Institute|volume=348|issue=8|pages=1797–1813|doi=10.1016/j.jfranklin.2011.05.001}}{{Cite book|title=Handbook of Geometric Computing|last1=Mao|first1=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}}{{Cite journal|last1=Behnia|first1=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|journal=Chaos, Solitons & Fractals|volume=35|issue=2|pages=408–419|doi=10.1016/j.chaos.2006.05.011|bibcode=2008CSF....35..408B}}
- Asymmetric chaos cryptography, where one key of the cryptosystem is public. Some of the few proposed systems {{Cite journal|date=2004-11-15|title=Public-key encryption with chaos|journal=Chaos: An Interdisciplinary Journal of Nonlinear Science|volume=14|issue=4|pages=1078–1082|doi=10.1063/1.1821671|pmid=15568922|issn=1054-1500|last1=Kocarev|first1=Ljupco|last2=Sterjev|first2=Marjan|last3=Fekete|first3=Attila|last4=Vattay|first4=Gabor|bibcode=2004Chaos..14.1078K}}{{Cite journal|last1=Kocarev|first1=L.|last2=Makraduli|first2=J.|last3=Amato|first3=P.|date=2005-10-01|title=Public-Key Encryption Based on Chebyshev Polynomials|journal=Circuits, Systems and Signal Processing|language=en|volume=24|issue=5|pages=497–517|doi=10.1007/s00034-005-2403-x|s2cid=123533966|issn=0278-081X}} have been broken.{{cite journal |last1=Bergamo |first1=P. |last2=D'Arco |first2=P. |last3=De Santis |first3=A. |last4=Kocarev |first4=L. |title=Security of public-key cryptosystems based on Chebyshev polynomials |journal=IEEE Transactions on Circuits and Systems I |date=July 2005 |volume=52 |issue=7 |pages=1382–1393 |doi=10.1109/TCSI.2005.851701 |arxiv=cs/0411030 |s2cid=18342884 |url=https://doi.org/10.1109/TCSI.2005.851701 |access-date=29 October 2021}}
The majority of chaos-based cryptographic algorithms are symmetric. Many use discrete chaotic maps in their process.{{Cite journal|last1=Behnia|first1=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|journal=International Journal of Bifurcation and Chaos|volume=18|issue=1|pages=251–261|doi=10.1142/S0218127408020288|issn=0218-1274|bibcode=2008IJBC...18..251B|arxiv=nlin/0601051|s2cid=9089024}}
Applications
= Image encryption =
Bourbakis and Alexopoulos{{Cite journal|last1=Bourbakis|first1=N.|last2=Alexopoulos|first2=C.|title=Picture data encryption using scan patterns|journal=Pattern Recognition|language=en|volume=25|issue=6|pages=567–581|doi=10.1016/0031-3203(92)90074-s|year=1992|bibcode=1992PatRe..25..567B}} 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.{{Cite journal|last1=Alvarez|first1=Gonzalo|last2=Li|first2=Shujun|date=2006-08-01|title=Some basic cryptographic requirements for chaos-based cryptosystems|journal=International Journal of Bifurcation and Chaos|volume=16|issue=8|pages=2129–2151|doi=10.1142/S0218127406015970|issn=0218-1274|bibcode=2006IJBC...16.2129A|arxiv=nlin/0311039|s2cid=222179832 }} 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.{{Cite journal|last1=Behnia|first1=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|journal=Physics Letters A|volume=366|issue=4–5|pages=391–396|doi=10.1016/j.physleta.2007.01.081|bibcode=2007PhLA..366..391B}}{{Cite journal|last1=Ghebleh|first1=M.|last2=Kanso|first2=A.|date=2014-06-01|title=A robust chaotic algorithm for digital image steganography|journal=Communications in Nonlinear Science and Numerical Simulation|volume=19|issue=6|pages=1898–1907|doi=10.1016/j.cnsns.2013.10.014|bibcode=2014CNSNS..19.1898G}}{{Cite journal|last1=Liu|first1=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|journal=Communications in Nonlinear Science and Numerical Simulation|volume=20|issue=2|pages=506–515|doi=10.1016/j.cnsns.2014.06.005|bibcode=2015CNSNS..20..506L}}{{Cite journal|last1=Behnia|first1=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|journal=Chaos, Solitons & Fractals|volume=40|issue=1|pages=505–519|doi=10.1016/j.chaos.2007.08.013|bibcode=2009CSF....40..505B|arxiv=0705.2633|s2cid=120158218}}{{Cite journal|last1=Kanso|first1=A.|last2=Ghebleh|first2=M.|date=2015-07-01|title=An efficient and robust image encryption scheme for medical applications|journal=Communications in Nonlinear Science and Numerical Simulation|volume=24|issue=1–3|pages=98–116|doi=10.1016/j.cnsns.2014.12.005|bibcode=2015CNSNS..24...98K}} 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.{{Cite journal|last1=Kwok|first1=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|journal=Chaos, Solitons & Fractals|volume=32|issue=4|pages=1518–1529|doi=10.1016/j.chaos.2005.11.090|bibcode=2007CSF....32.1518K}} 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.{{cite journal |last1=Youssef |first1=Mohammed |last2=Gabr |first2=Mohamed |last3=Alexan |first3=Wassim |last4=Marvy |first4=Mansour |last5=Kamal |first5=Karim |last6=Hosny |first6=Khalid |last7=El-Damak |first7=Dina |title=Enhancing Satellite Image Security Through Multiple Image Encryption Via Hyperchaos, SVD, RC5, and Dynamic S-Box Generation |journal=IEEE Access |date=2024 |volume=12 |page=123921-123945|doi=10.1109/ACCESS.2024.3454512 |doi-access=free }}{{Cite book|last1=Akhavan|first1=Amir|last2=Mahmodi|first2=Hadi|last3=Akhshani|first3=Afshin|title=Computer and Information Sciences – ISCIS 2006 |chapter=A New Image Encryption Algorithm Based on One-Dimensional Polynomial Chaotic Maps |date=2006-11-01|volume=4263|language=en|publisher=Springer, Berlin, Heidelberg|pages=963–971|doi=10.1007/11902140_100|series=Lecture Notes in Computer Science|isbn=978-3-540-47242-1}}{{Cite book|last1=Akhshani|first1=A.|last2=Mahmodi|first2=H.|last3=Akhavan|first3=A.|title=2006 International Conference on Image Processing |chapter=A Novel Block Cipher Based on Hierarchy of One-Dimensional Composition Chaotic Maps |date=2006-10-01|pages=1993–1996|doi=10.1109/ICIP.2006.312889|isbn=978-1-4244-0480-3|s2cid=14013123}}{{Cite book|last1=Chuanmu|first1=Li|last2=Lianxi|first2=H.|title=2007 International Workshop on Anti-Counterfeiting, Security and Identification (ASID) |chapter=A New Image Encryption Scheme based on Hyperchaotic Sequences |date=2007-04-01|pages=237–240|doi=10.1109/IWASID.2007.373734|isbn=978-1-4244-1035-4|s2cid=16930243}}
= Hash function =
Chaotic behavior can generate hash functions{{citation needed|date=September 2021}}.
= Random number generation =
The unpredictable behavior of the chaotic maps can be used in the generation of random numbers. Some of the earliest chaos-based random number generators tried to directly generate random numbers from the logistic map. Many more recent works did so using the numerical solutions of hyperchaotic systems of differential equations, either at the integer-order, or the fractional-order.{{cite journal |last1=Gabr |first1=Mohamed |last2=Elias |first2=Rimon |last3=Papakostas |first3=George |last4=Alexan |first4=Wassim |title=Image Encryption via Base-n PRNGs and Parallel Base-n S-Boxes |journal=IEEE Access |date=2023 |volume=11 |page=85002-85030|doi=10.1109/ACCESS.2023.3301460 |bibcode=2023IEEEA..1185002G |doi-access=free }}