CHAOS CRYPTOGRAPHYNathaniel SpeiserPhysics 330-2

What is Chaos Cryptography?• Simply put, the application of chaos theory, usually in the

form of chaotic maps, to cryptography

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

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…

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

Symmetric Key Cryptography• Alice and Bob use a key to encrypt/decrypt data that only

they know (hopefully)

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

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

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

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

Example of a chaos cryptosystem• Chebyshev polynomial map: Tp+1(x) = 2xTp(x)-Tp-1(x)

• Source of chaotic dynamics• Key generation algorithm:

1. Generate large integer s2. Select a random number x from [-1,1], compute Ts(x)3. Alice sets her public key to (x,Ts(x)) and her private key to s

• Encryption algorithm:1. Represent message as number M [-1,1]2. Generate large integer r3. Compute Tr(x), Trs(x) = Tr(Ts(x)), and X = M * Trs(x)4. Send ciphertext C = (Tr(x), X) to Alice

• Decryption Algorithm:1. Use private key s to compute Tsr(x) = Ts(Tr(x))2. Recover M by computing M = X/Tsr(x)

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

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

Sources• Kocarev, L., and Shiguo Lian. Chaos-based Cryptography.

Berlin:Springer, 2011. Print.• 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. <https://eprint.iacr.org/2012/263.pdf>.