$$e$$-CM: A novel approach to advancing chaotic dynamics in discrete one-dimensional maps for secure IoT applications

Abstract One-dimensional (1D) chaotic maps play a pivotal role in diverse fields, including control systems, secure communication, and modeling complex systems, enabling to capture and analyze the intricate behaviors of dynamical processes. However, their limited control parameter ranges constrain their effectiveness in security applications such as cryptography and pseudo-random number generation. Addressing these concerns, the paper proposes a universal $$e$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>e</mml:mi> </mml:math> -CM chaotification model, based on Euler’s number, to enhance the chaotic control parameter range of one-dimensional maps to infinity. The efficacy of the $$e$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>e</mml:mi> </mml:math> -CM model has been assessed for thirteen 1D chaotic maps including, Chebyshev, Logistic, Sine, Cubic, Coupled Sine, Cubic Logistic, Quadratic, Renyi, Simple Quadratic, Sine-Sinh-Sine, Singer, Squared Sine Logistic, and Tent maps. Various tests have been conducted to evaluate the enhanced $$e$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>e</mml:mi> </mml:math> -CM maps, such as Lyapunov exponents, bifurcation diagrams, approximate and sample entropies, time sensitivity analysis, 0–1 test, 2D and 3D phase plots, and cobweb plots. The findings indicate that $$e$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>e</mml:mi> </mml:math> -CM-based maps exhibit complex chaotic dynamics characterized by elevated positive Lyapunov exponent values, the lack of periodic windows in the bifurcation plot, and higher sample and approximate entropy values. The 2D and 3D phase trajectory plots demonstrate that the points are evenly distributed across the entire phase space, while the cobweb plots illustrate densely packed irregular rectangular trajectories. The 0–1 test produces linear $$M-t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>-</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> </mml:math> and Brownian motion resembling chaotic $$p-q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>-</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:math> plots, alongside indicator value approaching the ideal value of $$1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . Additionally, the $$e$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>e</mml:mi> </mml:math> -CM-based enhanced 1D maps have been employed in designing a pseudo-random bit generator (PRBG). The PRBG has been evaluated in the MATLAB simulator for operational efficiency, bit-generation speed, and performance, alongside a rigorous assessment of its statistical randomness through the NIST tests. The PRBG has proven to be efficient in resource usage, faster in execution, and high-performing, while also successfully passing NIST tests. The PRBG has also been implemented and analyzed on the leading ARM Cortex-M $$4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> -based LPC $$4357$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>4357</mml:mn> </mml:mrow> </mml:math> IoT development board to showcase its practical efficiency in RAM and ROM memory usage, execution time, current, energy, and power consumption. The results indicate that the PRBG demonstrates smaller memory usage, significantly reduced execution time, and minimal power and energy consumption. Thus, the $$e$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>e</mml:mi> </mml:math> -CM based enhanced maps and PRBG stand out as an excellent choice for use in IoT environments, where security, efficiency, and performance are paramount.