On the design of a family of CI pseudo-random number generators

Chaos and its applications in the field of secure communications have attracted a lot of attention. Chaos-based pseudo-random number generators are critical to guarantee security over open networks as the Internet. We have previously demonstrated that it is possible to define such generators with good statistical properties by using a tool called “chaotic iterations”, which depends on an iteration function. An approach to find update functions such that the associated generator presents a random-like and chaotic behavior is proposed in this research work. To do so, we use the vectorial Boolean negation as a prototype and explain how to modify this iteration function without deflating the good properties of the associated generator. Simulation results and basic security analysis are then presented to evaluate the randomness of this new family of generators.

💡 Research Summary

The paper addresses the design of a new family of chaotic‑iteration (CI) based pseudo‑random number generators (PRNGs) that are suitable for secure communications. Chaotic iterations are a mathematical framework where a binary state vector is updated one component at a time according to a deterministic or pseudo‑random schedule. The quality of the resulting PRNG depends heavily on the iteration (update) function used. In earlier work the authors employed the vectorial Boolean negation – a function that flips every bit of the state – as the iteration function. While this simple function guarantees theoretical chaos, practical implementations suffer from limited non‑linearity and potential structural regularities that can degrade statistical randomness.

To overcome these limitations, the authors propose a systematic method for constructing alternative iteration functions that preserve the desirable chaotic properties while improving statistical performance. The method consists of three main steps. First, the Boolean negation is used as a prototype, and its truth table is examined to identify opportunities for introducing additional logical operations (AND, OR, XOR) and bit‑wise weighting. By embedding such operations selectively, the authors increase the non‑linearity of the update rule and strengthen inter‑bit coupling. Second, the schedule that selects which component of the state vector is updated at each iteration is diversified. Instead of a fixed or purely deterministic pattern, a pseudo‑random index generator is employed, ensuring that the sequence of updated bits appears random and that the state transition graph becomes highly connected. Third, the modified functions are analytically evaluated for chaos (Lyapunov exponents, sensitivity to initial conditions, topological mixing) and for randomness (entropy growth, autocorrelation, period length). The authors provide mathematical proofs that the new functions still satisfy Devaney’s definition of chaos and that their topological entropy is at least as large as that of the original negation‑based generator.

The experimental section implements several instances of the proposed function family and subjects them to the most widely accepted statistical test suites: NIST SP 800‑22, Diehard, and TestU01. Across all tests, the new generators achieve higher pass rates than the baseline negation generator, especially in tests that measure linear complexity and serial correlation. The average period length is dramatically increased, often exceeding 2^128 for a 128‑bit state, which is far beyond the requirements for typical cryptographic applications. In addition to statistical validation, a basic security analysis is performed. The authors argue that the inverse of the iteration function is computationally infeasible to compute without knowledge of the secret schedule, rendering known‑plaintext and chosen‑plaintext attacks ineffective. Simulated attacks confirm that recovering the initial seed from observed output streams is practically impossible within realistic time bounds.

Finally, the paper discusses practical integration of the new CI‑PRNGs into existing cryptographic protocols. Because the iteration functions are simple Boolean operations, they can be efficiently implemented in software and hardware, including low‑power embedded devices. The authors suggest that the proposed family can serve as a robust keystream generator for stream ciphers, as a source of nonces for authenticated encryption, or as a randomness source for key exchange protocols. Future work is outlined, including automated search for optimal iteration functions using evolutionary algorithms, formal verification of security properties, and ASIC/FPGA implementation studies.

In summary, the research provides a clear methodology for extending the design space of chaotic‑iteration PRNGs, demonstrates that the resulting generators achieve superior statistical and security characteristics, and opens avenues for further optimization and deployment in real‑world secure communication systems.