# Chaotic Cryptography: Applications of Chaos Theory to Cryptography

### Abstract:

Chaos theory is the study of dynamical systems that are highly sensitive to initial conditions. These "chaotic" functions exhibit seeming randomness given an initial value, or seed. From a cryptographic standpoint, having a determinsitic algorithm for creating true randomness is a very appealing concept.

In recent years, work has been done to try and create some cryptographic ciphers based upon chaotic functions, such as the logistic map. In this paper, I will discuss the motivation behind applying chaos theory to cryptography, discuss the work done thus far on the subject, and analyze proposed cryptographic functions based upon the Logistic Map, a chaotic function. I will write implementations in Python, and will be use NIST's statistical test suite in order to examine the randomness obtained from the output of these algorithms.
### Outline of Paper

- An Overview of Chaotic Dynamics
- History of Chaotic Cryptography
- Implementing Chaotic Ciphers
- Statistical Analysis
- Frequency (Monobit) Test
- Run Test
- Binary Rank Test
- Spectral (FFT) Test
- Linear Complexity Test
- Entropy Test

- Conclusions
- Limitations
- Potential Future Improvements

### References:

Gutowitz, H. (1993). Cryptography with Dynamical Systems. *Cellular Automata and Cooperative Systems*, 237-274. doi:10.1007/978-94-011-1691-6_21

Borujeni, Shahram Etemadi, and Mohammad Saeed Ehsani. "Modified Logistic Maps for Cryptographic Application." *Applied Mathematics* 06.05 (2015): 773-82. Web.

Arroyo, David, Gonzalo Alvarez, and Veronica Fernandez. "On the Inadequacy of the Logistic Map for Cryptographic Applications." *Internet Archive*. Universidad De Salamanca, 28 May 2008. Web. 22 Mar. 2017.

Goyal, Yamini, Geet Kalani, and Shreya Sharma. "2-Step Logistic Map Chaotic Cryptography Using Dynamic Look-up Table." *International Journal of Computer Applications*, 10 Nov. 2015. Web. 22 Mar. 2017.

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