On the use of dynamical systems in cryptography

TL;DR

The goal of this paper is to cast discrete dynamical systems problems in a modern cryptographic and complexity theoretic language, so that researchers working in chaos-based cryptography can begin designing cryptographic protocols that have a better chance of meeting the extreme standards of modern cryptography.

Abstract

Ever since the link between nonlinear science and cryptography became apparent, the problem of applying chaotic dynamics to the construction of cryptographic systems has gained a broad audience and has been the subject of thousands of papers. Yet, the field has not found its place in mainstream cryptography, largely due to persistent weaknesses in the presented systems. The goal of this paper is to help remedy this problem in two ways. The first is by providing a new algorithm that can be used to attack -- and hence test the security of -- stream ciphers based on the iteration of a chaotic map of the interval. The second is to cast discrete dynamical systems problems in a modern cryptographic and complexity theoretic language, so that researchers working in chaos-based cryptography can begin designing cryptographic protocols that have a better chance of meeting the extreme standards of modern cryptography.

Figures (1)

• Figure 1: The graph on the top left represents the first iteration of the tent map, with the black intervals representing the points $A^{(1)}$ of $[0,1]$ such that $g(f[A^{(1)}]) = c_1 = 0$. The graph on the top right represents the second iteration of the tent map, $f^2$, with the black intervals representing the subset $A^{(2)}$ of $A^{(1)}$ such that $g(f^2[A^{(2)}]) = c_2 = 1$. Similarly, the bottom left and right graphs represents the third and fourth iterates of the tent map with the black intervals representing the subsets $A^{(3)}, A^{(4)}$ with $A^{(4)} \subset A^{(3)} \subset A^{(2)} \subset A^{(1)}$ such that $g(f^3[A^{(3)}]) = c_3 =0$ and $g(f^4[A^{(4)}]) = c_4 = 0$. Note that the length of the black intervals representing candidate secret initial conditions is halved with each iteration of the algorithm.

Theorems & Definitions (11)

• Definition 2.1
• Definition 2.2
• Definition 2.3: Pseudorandom generator (PRG)
• Definition 2.4: One-way functions (OWF)
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6