Chaos Theory and Encryption

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Chaos Theory and Encryption

• Jeffrey L. Duffany
• Universidad del Turabo
• School of Engineering
• Department of Electrical Engineering

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Chaos Theory

• A name given to wide-ranging attempts to uncover
  the statistical regularity hidden in processes
  that otherwise appear random.
• Applied to diverse phenomena such as turbulence
  in fluids, weather patterns, motion in energy
  fields predator-prey cycles, the spread of
  disease, and even the onset of war.

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Hurricane Isabela September 2003
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Chaos in Mathematics

• Some simple mathematical equations exhibit
  complex behavior which has been called chaotic
• Difference/differential equations
• Recursion
• Nonlinearities
• Newtons Method with complex roots

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The Mandelbrot Setz z2c
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The Mandelbrot Setz z2c
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Chaos Theory

• Systems described as "chaotic" are extremely
  susceptible to changes in initial conditions.
• As a result, small uncertainties in measurement
  are magnified over time, making chaotic systems
  predictable in principle but unpredictable in
  practice.

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Encryption Algorithms

• PermutationPermutation is a kind of diffusion.
  This technique is a simple rearrangement of the
  letters of plain text (coffee -gt eeffoc)
• SubstitutionSubstitution is a kind of confusion.
  This technique is to substitute one character
  into the other (ibmhal).

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Uses of Encryption

• Credit-card information
• Social Security numbers
• Private correspondence
• Sensitive company information
• Bank-account information

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Characteristics of Encryption Algorithms

• Encryption algorithms use complex formula and
  large key values for encrypting, including 40-bit
  or even 128-bit numbers.
• A 128-bit number has a possible 2128 or
  3,402,823,669,209,384,634,633,746,074,300,000,000,
  000,000,000,000,000,000,000,000,000,000 different
  combinations.

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The Goals of Encryption

• To provide an easy and inexpensive means of
  encryption and decryption to all authorized users
  in possession of the key
• To make it difficult and/or expensive to find the
  plain text without the use of the key.

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Classical Encryption - Disadvantage

• Techniques well known and understood
• Amount of time for encoding decoding can increase
  significantly with the size of the key
• Same sequence is always encoded the same way
  which can vulnerability to cryptanalysis

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Chaotic Encryption

• Based on mathematical formula which exhibit
  chaotic behavior
• For example the population growth a.k.a. Logistic
  Map xrx(1-x)
• The key for the method is the choice of r and x

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Solution to Logistic Map Equationxrx(x-1)
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General Chaotic Encryption Method

• Baptista, M. S. (1998 March 16).
  Cryptography with chaos. Physics Letters A, 240
  (1-2), 50-54.

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General Chaotic Encryption Method

• Choose key (r,x)
• Map symbol set (A,B,C) e.g. (.49ltTlt.51)
• Choose first symbol to send (e.g. T)
• Iterate formula x rx(1-x) n times until x
  enters T space (for example .49ltTlt.51)
• Send n as coded version of symbol

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General Chaotic Encryption Method

• To Decode
• Set key parameters (r,x)
• Receive n
• Iterate formula x rx(1-x) n times
• Determine symbol (T)

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General Chaotic Encryption Method

• Variation
• Choose key (r,x)
• Map symbol set (A,B,C) e.g. (.49ltTlt.51)
• Choose first symbol to send (e.g. T)
• Generate a random number k
• Iterate formula x rx(1-x) n times until x
  enters T space for kth time (for example
  .49ltTlt.51)
• Send n as coded version of symbol

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Inherent Property of General Chaotic Encryption
Method

• Any given symbol such as T will may be
  given as a different code each time. For example,
  suppose k is a random number between 1 and 10
• K 1 T 511
• K 3 T 3339
• K 9 T 12345
• K 3 T 3339

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Inherent Property of General Chaotic Encryption
Method

• A given symbol such as T will be sent as a
  different code each time.
• The sender does not have to send the number k
  to the receiver.
• As illustrated in the following four diagrams the
  character frequency of a scrambled and
  unscrambled file appear indistinguishable

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Unscrambled file character frequency
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Scrambled File character frequency

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Typical file(encrypted) Character frequency
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Scrambled File (encrypted) character frequency
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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