Seminar on Chaos Based Cryptography

Chaos based
Cryptography
Presented by:
Amirul Haque Muhammad Hamid (16MCS001)
(16MCS012)

Introduction
• Chaos refers to the complex, difficult-to-predict behavior
found in nonlinear systems.
Chaos : the state of randomness or confusion
• Chaos theory: A field of study in mathematic
• Study the behavior of dynamic system with initial condition
• Sensitive dependence on initial conditions means that a very
small change in the initial state of a system can have a large
effect on its later state.

Characteristics of chaos
• They are aperiodic.
• They exhibit sensitive dependence on initial conditions and
unpredictable in the long term.
• They are governed by one or more control parameters, a
small change in which can cause the chaos to appear or
disappear.
• Their governing equations are nonlinear.
• Chaotic system will produce the same results if given the same
inputs, it is unpredictable in the sense that you can not predict
in what way the system's behavior will change for any change
in the input to that system. a random system will produce
different results when given the same inputs.

Types of system
• All systems can be basically divided into three types:
•
• 1.Deterministic systems
• These are systems for which for a given set of conditions the
result can be predicted and the output does not vary much with change in initial
conditions.
•
• 2. Stochastic systems
• These systems, which are not as reliable as deterministic
systems. Their output can be predicted only for a certain range of values.
•
• 3. Chaotic systems
• Chaotic systems are the most unpredictable of the three
systems. Moreover they are very sensitive to initial conditions and a small
change in initial conditions can bring about a great change in its output.
•

Example of Chaotic
System
• Chaos theory offers ordered models for seemingly
disorderly systems, such as:
 Weather patterns
 Turbulent Flow
 Population dynamics
 Stock Market Behavior
 Traffic Flow
 Nonlinear circuits

Bifurcation
• The logistic map shows a variety of behaviors and it has
transitions between these behaviors as we change the
parameter r. Such transitions in dynamical systems are called
bifurcations.
• Bifurcation is a scientific way to say something splits in
two—branches.
• If patterns bifurcate quickly enough, they can become
complex very fast, leading to bifurcation cascade and chaos.

Lyapunov exponent
How to quantify chaos?
0
)(
ln
1
lim
d
td
tt 

d0
d(t)
The Lyapunov exponent characterizes the rate of exponential divergence of
nearby orbits
It is formally defined as:
The rate of divergence may depend on the orientation of the d0 vector

Cont..
• Lyapunov exponents are defined as the long time average
exponential rates of divergence of nearby states.
• If a system has at least one positive Lyapunov exponent, than the
system is chaotic.
• The larger the positive exponent, the more chaotic the system
becomeGives a measure for the predictability of a dynamic system
• Describes the average rate which predictability is lost
• Usually Calculate the Maximal Lyapunov Exponent
• Gives the best indication of predictability
• Positive value usually taken as an indication that the system is
chaotic

Information Entropy
• The information entropy is design to evaluate the
uncertainty in a random variable.
• The evaluation equation is
• The information entropy has a maximum when all signal
values have random distributions.

Chaos Based Cryptography
• Chaotic cryptology = chaotic cryptography + chaotic cryptanalysis
• Cryptography: The art of hiding messages
• Cryptanalysis: The art of decrypting or obtaining plain text
from hidden messages

Why someone choose chaos based cryptosystem over
traditional cryptosystem
• Traditional symmetric ciphers such as Advanced Encryption
Standard (AES) are designed with good confusion and diffusion
properties.
• These two properties can also be found in chaotic systems which
have desirable properties of pseudo-randomness, ergodicity, high
sensitivity to initial conditions and parameters.
• Chaotic maps have demonstrated great potential for information
security, especially image encryption, while the standard
encryption methods as the AES algorithm seem not to be suitable to
cipher such type of data.

Chaotic cryptography and standard algorithm
Similarities and differences between chaotic systems and cryptographic algorithms.
Highly secured
and
fast execution

How to design chaotic cryptosystem
Figure. A procedure for a design of a chaos-
based block-encryption algorithm.

Encryption Algorithm
The encryption process of the proposed algorithm [1]

Experimental Results of above algorithm
(a) Original image, (b-c-d) Histograms of the R, G, B components of the original image,
(e) Encrypted image, (f-g-h) Histograms of the R, G, B components of the encrypted image

Decrypting above encrypted image
(a) Encrypted image (Sailboat on lake), (b) Decrypted image by using K, (c) Decrypted image by using K1

Example 2:
Pseudo Random generator using chaotic system [2]
• Random numbers are mainly used to create
secret keys or random sequences.
• This Pseudo random number generator (PRNG) based
on chaotic maps and S-Box tables.
•

Block of the PRNG
Block diagram of the PRNG

Performance and Security Analysis
• Histogram Analysis
• Speed Analysis
Histogram of 43 000 numbers

Performance and Security Analysis
• Randomness Tests
Since the computed P-value of each test is > 0.01, then
conclude that the output sequence of our PRNG is random

Chaos Based Secure Hash
• The chaotic hash function is new trend in cryptography.
• Hashing using chaotic system needs defining the
mapping scheme for trajectory, choosing valid initial
condition and parameters
Diagram of chaotic hash function[3]

Detail construction of CBHF[3]

Application of chaos in
Cryptography
• Block cipher
• Pseudo random number generation
• Public Key algorithm
• Chaotic Communication
• Chaos in image encryption
• Chaotic Neuronal Networks
• Genetic networks
• Design of Chaotic Circuit

Summary
• Chaotic encryption not as well known as standard
encryption methods (e.g.,DES).
• Applicable to a wide range of encryption techniques –
e.g. chaotic masking.
• Potential to be as strong as other existing methods
• Potential to be easier to compute – eliminate need for file
scrambling
• Potentially less vulnerable to cryptanalysis

References
[1] R. Parvaz, M. Zarebnia, “A combination chaotic system and application in color image
encryption,” Optics and Laser Technology, Vol. 101, PP 30–41, (2018)
[1] Chanil Pak, Lilian Huang, A new color image encryption using combination of 1D chaotic
map, Signal Process.138 (2017) 129–137.
[3] Mohamed Amin, Osama S. Faragallah, Ahmed A. Abd El-Latif, “Chaos-based hash function
(CBHF) for cryptographic applications”, Chaos, Solitons and Fractals 42 (2012) 767–772.
[2] Nigel Crook and Tjeerd olde Scheper, A Novel Chaotic Neural Network Architecture
[3] Chaos and Time-Series Analysis, by J.C. Sprott, Oxford Press 2006
[4] S. M. Chang, M. C. Li and W. W. Lin, Asymptotic synchronization of modified logistic
hyper-chaotic systems and its applications. Nonlinear Analysis: Real World Applications, Vol.
10, Issue 2 (2009), pp. 869–880.
[5] Nigel Crook and Tjeerd olde Scheper, A Novel Chaotic Neural Network Architecture
[6] R. Parvaz ⇑, M. Zarebnia

Seminar on Chaos Based Cryptography

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