1. CHAOS CRYPTOGRAPHY Nathaniel Speiser Physics 330-2

2. What is Chaos Cryptography? Simply put, the application of chaos theory, usually in the form of chaotic maps, to cryptography

3. What is Cryptography? A method of storing and transmitting data in a particular form so only certain entities can read and process it Key words: Plaintext: the message being transmitted in its original form Ciphertext: the message that actually is transmitted Cipher: the pair of algorithms that encrypt/decrypt the message Key: what the cipher uses to encrypt/decrypt the message Obviously very important in the modern information age

4. A brief history Dates back to the Antiquity Until the 1400s and the invention of the Vigenere cypher, all cryptosystems were relatively simple Alphabet shifting, substitution, simple cryptographic devices Cryptography advanced slowly until…

5. Modern Cryptography With the advent of computers there was vastly more information being transmitted so more data security was needed Modern Cryptography based heavily in information theory and modern mathematics

6. Symmetric Key Cryptography Alice and Bob use a key to encrypt/decrypt data that only they know (hopefully)

7. Asymmetric key cryptography Bob uses a public key to encrypt his message, Alice uses her (mathematically related) private key to decrypt it Often based on the computational complexity of certain problems Example: RSA based on difficulty of integer factorization

8. How does this relate to Chaos? Two key concepts in cryptography: diffusion and confusion Confusion: The key does not relate in a simple way to the ciphertext – why complex mathematical techniques are used Diffusion: if a character in the plaintext is changed, then several characters in the ciphertext should change, and vice versa

9. Close relation to Chaos concepts Confusion  Seemingly random behavior in chaotic systems Also related to ergodicity of chaotic systems Diffusion  Sensitivity to initial conditions Other similarities: Deterministic process causes pseudo-random behavior Simple processes have high complexity

10. 3 main applications of chaos in cryptography Block ciphers: transform short strings into a string of the same length with a secret key Pseudo-random number generation: many cryptographic algorithms require at least seemingly random numbers Public key algorithms: using chaotic maps in previously discussed cryptosystems

11. Example of a chaos cryptosystem Chebyshev polynomial map: T p+1 (x) = 2xT p (x)-T p-1 (x) Source of chaotic dynamics Key generation algorithm: 1. Generate large integer s 2. Select a random number x from [-1,1], compute T s (x) 3. Alice sets her public key to (x,T s (x)) and her private key to s Encryption algorithm: 1. Represent message as number M [-1,1] 2. Generate large integer r 3. Compute T r (x), T rs (x) = T r (T s (x)), and X = M * T rs (x) 4. Send ciphertext C = (T r (x), X) to Alice Decryption Algorithm: 1. Use private key s to compute T sr (x) = T s (T r (x)) 2. Recover M by computing M = X/T sr (x)

12. Problems with using chaos Cryptographic scheme from last slide is easily broken Generally algorithms using chaos are slower than conventional ones Unpredictability of chaos only comes out in long term – result of chaotic systems being continuous One main advantage - an infinite number of chaotic algorithms can be invented

13. Research Proposal Investigate more successful uses of chaos cryptography (image encryption, PRNGs, etc.) Research more on relationship between chaos and current types of encryption algorithms

14. Sources Kocarev, L., and Shiguo Lian. Chaos-based Cryptography. Berlin:Springer, 2011. Print. http://konwersatorium.pw.edu.pl/wyklady/2010_VLZ7_02_ wyklad.pdf http://konwersatorium.pw.edu.pl/wyklady/2010_VLZ7_02_ wyklad.pdf Cheong, Kai-Yuen. "One-way Functions from Chebyshev Polynomials." (2012): n. pag. Web. 10 Feb. 2016..