Chaos Theory and Encryption

Jeffrey L. Duffany

Universidad del Turabo School of Engineering Department of Electrical Engineering

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.

Hurricane Isabela – September 2003

Chaos in Mathematics

 Some simple mathematical equations exhibit complex behavior which has been called chaotic  Difference/differential equations    Recursion Nonlinearities Newton’s Method with complex roots

The Mandelbrot Set z = z**2+c

The Mandelbrot Set z = z**2+c

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.

Encryption Algorithms



Permutation

Permutation is a kind of diffusion. This technique is a simple rearrangement of the letters of plain text (coffee -> eeffoc) 

Substitution

Substitution is a kind of confusion. This technique is to substitute one character into the other (ibm=hal).

Uses of Encryption

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

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 2 128 or 3,402,823,669,209,384,634,633,746,074,300, 000,000,000,000,000,000,000,000,000,000,00 0,000 different combinations.

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.

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

Chaotic Encryption

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

Solution to Logistic Map Equation x=r*x*(x-1)

General Chaotic Encryption Method

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

General Chaotic Encryption Method

Choose key (r,x)  Map symbol set (A,B,C…) e.g. (.49

General Chaotic Encryption Method

To Decode:     Set key parameters = (r,x) Receive n Iterate formula x = r*x*(1-x) n times Determine symbol (=T)

General Chaotic Encryption Method

Variation:       Choose key (r,x) Map symbol set (A,B,C…) e.g. (.49

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

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

Unscrambled file – character frequency

Scrambled File – character frequency

Typical file (encrypted) – Character frequency

Scrambled File (encrypted) – character frequency

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