Cryptographically Secure Pseudo-Random Number Generation (CS-PRNG) Design using Robust Chaotic Tent Map (RCTM)

TL;DR

This work introduces a cryptographically secure PRNG based on a Robust Chaotic Tent Map (RCTM) that extends chaos across a wide parameter range $\mu\in(2,100)$ using modulo and scaling operations. By thresholding the chaotic trajectory with $\tau=0.5$, the method yields binary streams whose randomness is validated through NIST STS, ENT, and TestU01, alongside key-space and sensitivity analyses. The approach demonstrates large key-space, low sequence correlation, high information entropy, and strong differential properties, indicating practical suitability for cryptographic services. Overall, the RCTM-based CS-PRNG offers robust chaotic behavior, extensive parameter flexibility, and strong empirical security evidence for secure randomness generation.

Abstract

Chaos, a nonlinear dynamical system, favors cryptography due to their inherent sensitive dependence on the initial condition, mixing, and ergodicity property. In recent years, the nonlinear behavior of chaotic maps has been utilized as a random source to generate pseudo-random number generation for cryptographic services. For chaotic maps having Robust chaos, dense, chaotic orbits exist for the range of parameter space the occurrence of chaotic attractors in some neighborhoods of parameter space and the absence of periodic windows. Thus, the robust chaotic map shows assertive chaotic behavior for larger parameters space with a positive Lyapunov exponent. This paper presents a novel method to generate cryptographically secure pseudo-random numbers (CSPRNG) using a robust chaotic tent map (RCTM). We proposed a new set of equations featuring modulo and scaling operators that achieve vast parameter space by keeping chaotic orbit globally stable and robust. The dynamic behavior of the RCTM is studied first by plotting the bifurcation diagram that shows chaotic behavior for different parameters, which the positive Lyapunov exponent verifies. We iterated the RCTM to generate pseudo-random bits using a simple thresholding method. Various statistical tests are performed that ascertain the randomness of generated secure pseudo-random bits. It includes NIST 800-22 test suite, ENT statistical test suite, TestU01 test suite, key space analysis, key sensitivity analysis, correlation analysis, histogram analysis, and differential analysis. The proposed scheme has achieved larger key space as compared with existing methods. The results show that the proposed PRBG algorithm can generate CSPRNG.

Figures (15)

• Figure 1: Bifurcation diagram of Classical Tent Map
• Figure 2: Lyapunov Exponent of Classical Tent Map
• Figure 3: RCTM with Scaling and modulo operation a) $\mu=2.75$ b) $\mu= 4.15$
• Figure 4: Correlation between the generated sequences of Tent Map and CTM with variable control parameter
• Figure 5: The plots illustrate the ergodic behavior of the CTM for different values of the control parameter $\mu$. The ergodicity is shown by the distribution of the values $x_n$ over iterations $n$. (a) $\mu = 1.2$ shows limited spread, indicating less ergodic behavior. (b) $\mu = 1.5$ demonstrates a wider spread. (c) $\mu = 1.9$ exhibits even more spread, and (d) $\mu = 1.99$ shows a near-uniform distribution, indicating highly ergodic behavior.