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Chaos Theory In Cryptography

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john marshall lazar

on 7 December 2012

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Transcript of Chaos Theory In Cryptography

Chaos Theory In Cryptography Rabbit Cipher Conclusion Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy.
Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the butterfly effect Chaos theory is applied in many scientific disciplines:
mathematics
programming
microbiology
biology
computer science
economics
engineering
finance
meteorology
philosophy
physics
politics
population dynamics
psychology
and robotics. The Butterfly Effect.
Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas? One of the characteristics of chaotic dynamics is a sensitivity to initial conditions; i.e., two relatively close initial values will diverge as the system evolves.
The chaotic behavior is too difficult to predict by analytical methods without the secret key being known.
This would reduce a potential attack to one category, that of a brute force attack, in which all possible keys are tested against the encrypted data Infact,
Bruce Schneier, a noted cryptologists, estimates it would take the energy output of most of the suns in our universe to power a computer to just count that high, let alone make a brute force attempt at decryption Advanced Cipher Rabbit cipher is a relatively new stream cipher that was inspired by the random behavior of chaotic maps.
Briefly speaking, it is constructed using a chaotic system of coupled non¬linear maps that exhibits secure cryptographic properties in its discretized form.
It is designed to work with 128-bit data blocks, as both the key and output data are 128 bits in length. Additionally, its internal data structure consists of eight state variables and eight counters. Overall, however, there are certainly elements of chaos theory that make it theoretically applicable to cryptography.
For this reason, there has been and will probably continue to be significant research done in chaos-based symmetric key cryptosystems.
However, given the loose connection between these two fields thus far, it is difficult to tell if these research efforts will be successful when compared to today’s standardized cryptographic primitives and the emerging usage of elliptic curves and other number theo¬retical concepts in cryptography.
Perhaps as chaos theory evolves this connection will become clearer and pave the way for more appropriate cryptography applications. Until then, however, traditional number theory cryptosystems will continue to lead the way into the future. Chaotic Fiestel Cipher Image Encryption Schemes Based on 2-D Chaotic Maps Assuming that the size of the plain-image is M x N, the encryption procedure can be described as follows (see Figure 4.3):
• define a discretized and invertible 2-D chaotic map on an M x N lattice, where the discretized parameters serve as the secret key;
• iterate the discretized 2-D chaotic maps on the plain-image to permute all pixels;
• use a substitution algorithm (cipher) so as to modify the values of all pixels to flatten the histogram of the image, i.e., to enable the confusion property;
• repeat the permutation and the substitution for k rounds to generate the cipher-image The most favorable 2-D chaotic map for discretization is the Baker map whose continuous form is defined as follows:
B(x, y) =
(2x,y/2),0 < x < 1/2,
(2x — 1, y/2 + 1/2), 1/2 < x < 1.
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