[Paper Review] Randomness and disorder of chaotic iterations. Applications in information security field

Xiaole Fang, Christophe Guyeux+2 more|arXiv (Cornell University)|Nov 25, 2016

Chaos-based Image/Signal Encryption5 references3 citations

TL;DR

This paper proposes a framework for leveraging chaotic iterations—discrete dynamical systems on integer domains—to design cryptographically secure primitives such as pseudorandom number generators, hash functions, and symmetric ciphers. By proving that these systems satisfy Devaney’s chaos criteria (sensitivity, transitivity, and dense periodic points), the authors demonstrate their strong randomness and disorder properties, enabling robust security applications resistant to finite-precision degradation inherent in real-domain chaotic systems.

ABSTRACT

Design and cryptanalysis of chaotic encryption schemes are major concerns to provide secured information systems. Pursuing our previous research works, some well-defined discrete chaotic iterations that satisfy the reputed Devaney's definition of chaos have been proposed. In this article, we summarize these contributions and propose applications in the fields of pseudorandom number generation, hash functions, and symmetric cryptography. For all these applications, the proofs of chaotic properties are outlined.

Motivation & Objective

• To address dynamic degradation in real-domain chaotic systems caused by finite-precision arithmetic in digital implementations.
• To establish a discrete, integer-based framework—chaotic iterations—that avoids finite-precision effects while preserving chaotic behavior.
• To apply chaotic iterations to practical information security primitives: pseudorandom number generation, cryptographic hashing, and symmetric encryption.
• To formally prove that the proposed constructions satisfy Devaney’s definition of chaos, ensuring sensitivity, transitivity, and dense periodic points.
• To demonstrate that chaos in the system translates into strong cryptographic properties such as collision resistance and second-preimage resistance in hash functions.

Proposed method

• Define chaotic iterations as a discrete dynamical system where only one component of a binary vector is updated at each step, based on a sequence $ S^n \in \llbracket 1, N \rrbracket $.
• Model the system as a map $ G_f $ on the phase space $ \mathcal{X} = \llbracket 1, N \rrbracket^\mathbb{N} \times \mathds{B}^N $, combining a shift function $ \sigma $ and a state update function $ F_f $.
• Prove topological transitivity and regularity (dense periodic points) in the metric space $ (\mathcal{X}, d) $, satisfying Devaney’s chaos definition.
• Construct a pseudorandom number generator (PRNG) by using a secure PRNG to generate the iteration sequence $ S^n $, ensuring chaotic evolution.
• Design a hash function $ \mathcal{H}_h $ by applying chaotic iterations with a strategy derived from the input stream and a secure compression function $ h $, proving that $ \mathcal{H}_h $ inherits collision and second-preimage resistance from $ h $.
• Reformulate the Cipher Block Chaining (CBC) mode as a chaotic dynamical system using the map $ G_{\varepsilon_k \circ F_{f_0}} $, and prove chaos if the associated graph $ \mathcal{G}_g $ is strongly connected.

Experimental results

Research questions

• RQ1 Can discrete chaotic iterations on integer domains provide a robust alternative to real-domain chaotic systems in cryptographic applications?
• RQ2 Do chaotic iterations satisfy Devaney’s formal definition of chaos, and can this be proven in a discrete, finite metric space?
• RQ3 Can chaotic iterations be used to construct secure pseudorandom number generators that avoid finite-precision degradation?
• RQ4 Does the application of chaotic iterations to hash functions preserve essential cryptographic properties like collision resistance and second-preimage resistance?
• RQ5 Is the standard CBC mode of operation inherently chaotic under Devaney’s criteria, and under what conditions does this hold?

Key findings

• Chaotic iterations defined on integer domains avoid finite-precision degradation, ensuring stable and unpredictable dynamics in digital implementations.
• The proposed PRNG based on chaotic iterations inherits chaotic behavior from the underlying system, ensuring strong randomness and sensitivity to initial conditions.
• The constructed hash function $ \mathcal{H}_h $ preserves the collision resistance and second-preimage resistance of the underlying compression function $ h $, provided $ h $ is secure.
• The CBC mode of operation is proven to be chaotic if the associated directed graph $ \mathcal{G}_g $ is strongly connected, linking classical cryptography with topological chaos.
• The system's chaotic dynamics are formally established via Devaney’s criteria: sensitivity, topological transitivity, and dense periodic points in the metric space.
• The framework enables the design of cryptographically secure primitives with provable dynamical chaos, offering a new theoretical foundation for chaos-based cryptography.