**1. Introduction**

Chaos theory has found numerous applications over the last 50 years, including, but not limited to, encryption, engineering, secure communications, robotics, biology, and economics—see, for example, Refs. [1–4] and the references cited therein. Nonlinear systems with chaotic properties are deterministic systems with high sensitivity to small changes in initial conditions and parameters which lead to completely different solution trajectories. This sensitivity, combined with their deterministic nature, makes chaotic systems a perfect basis for designs that require high complexity and increased security.

Due to their abovementioned usability, there is an ongoing demand for constructing novel chaotic systems. Moreover, it is of interest to develop minimal chaotic systems that can provide high performance when implemented on FPGA [5,6]. Most of constructed chaotic maps are modifications of known chaotic systems. These modifications usually follow simple techniques, such as introducing additional nonlinear terms in the system's differential/difference equations, changing an existing term to a higher-order term, or even by adding new variables to make the system hyperchaotic that is, having at least two positive Lyapunov exponents, which can only happen for four-dimensional systems or higher.

The logistic map [7,8] is one of the most well-known one-dimensional discrete time systems with chaotic behavior. Originally considered as a population model, it eventually found numerous applications in encryption, due to its simple and elegant form. This has also lead to many subsequent modifications of the map—see, for example, [9–17].

In this study, we consider a modification of the logistic map considered in [9]. The proposed modification is obtained by combining the map [9] with the conventional logistic map. This is done by multiplying the values of the map in [9] by the values of the logistic map computed from the decimal part of the same map, yielding a more complex behavior. The original map [9] exhibits a symmetric bifurcation diagram and constant chaos with a constant Lyapunov exponent, yet the proposed chaotic map showcases a plethora of chaos related phenomena, like antimonotonicity, crisis, and coexisting attractors, and the symmetric bifurcation diagram is now modified. In addition, the Lyapunov exponent of the new map can reach higher values, so overall the proposed map has a more complex chaotic behavior compared to the original map, and consequently compared to the classic logistic map as well. The emergence of many chaos related phenomena in the proposed system is an indication that the method of combining a given map with the logistic map derived from its decimal part can be generally utilized as a technique to make the behavior of a given system more complex.

Moreover, the proposed map is applied to the problem of pseudo-random bit generation [2,3, 10,11,13,18–36]. The term pseudo comes from the fact that a deterministic system is used to generate the sequence, rather than a random process, which is the case in true random bit generators. If the generator is properly designed though, the resulting sequence will have the characteristics of a random sequence. The method used utilizes various simple techniques like combinations of multiple maps, the comparison between different decimal parts of a number, bit reversal, and the *XOR* operator. The sequences obtained by the pseudo-random bit generator based on the new map passes all 15 of the NIST statistical tests, shows good correlation and cross-correlation characteristics, and has a satisfactory key space. Thus, it is suitable for encryption related applications.

The rest of the work is structured as follows: In Section 2, the proposed modification of the well-known logistic map is presented and studied. In Section 3, the considered chaotic system is applied to the problem of random bit generation. Finally, Section 4 concludes the work with a discussion on future research topics.
