1. Chaos Theory and Encryption Jeffrey L. Duffany Universidad del Turabo School of Engineering Department of Electrical Engineering

2. 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.

3. Hurricane Isabela – September 2003

4. 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

5. The Mandelbrot Setz = z**2+c

6. The Mandelbrot Setz = z**2+c

7. 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. Chaos Theory

8. 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). Encryption Algorithms

9. Credit-card information Social Security numbers Private correspondence Sensitive company information Bank-account information Uses of Encryption

10. 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. Characteristics of Encryption Algorithms

11. 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. The Goals of Encryption

12. 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 Classical Encryption - Disadvantage

13. 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 Chaotic Encryption

14. Solution to Logistic Map Equationx=r*x*(x-1)

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

16. 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

17. 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

18. 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 General Chaotic Encryption Method

19. 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

20. 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 Inherent Property of General Chaotic Encryption Method

21. Unscrambled file – character frequency

22. Scrambled File – character frequency

23. Typical file(encrypted) – Character frequency

24. Scrambled File (encrypted) – character frequency

25. 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 Summary