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 Setz = z**2+c

The Mandelbrot Setz = 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

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

SubstitutionSubstitution 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 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.

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 Equationx=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<T<.51) Choose first symbol to send (e.g. T) Iterate formula x = r*x*(1-x) n times until x

enters T space (for example .49<T<.51) Send n as coded version of symbol

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<T<.51) Choose first symbol to send (e.g. T) Generate a random number k Iterate formula x = r*x*(1-x) n times until x enters

T space for kth time (for example .49<T<.51) Send n as coded version of symbol

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