Because of the COVID-19 pandemic, most of the tasks have shifted to an online platform. Sectors such as e-commerce, sensitive multi-media transfer, online banking have skyrocketed. Because of this, there is an urgent need to develop highly secure algorithms which can not be hacked into by unauthorized users. The method which is the backbone for building encryption algorithms is the pseudo-random number generator based on chaotic maps. Chaotic maps are mathematical functions that generate a highly arbitrary pattern based on the initial seed value. This manuscript gives a summary of how the chaotic maps are used to generate pseudo-random numbers and perform multimedia encryption. After carefully analyzing all the recent literature, we found that the lowest correlation coefficient was 0.00006, which was achieved by Ikeda chaotic map. The highest entropy was 7.999995 bits per byte using the quantum chaotic map. The lowest execution time observed was 0.23 seconds with the Zaslavsky chaotic map and the highest data rate was 15.367 Mbits per second using a hyperchaotic map. Chaotic map-based pseudo-random number generation can be utilized in multi-media encryption, video-game animations, digital marketing, chaotic system simulation, chaotic missile systems, and other applications.
They have tested their number generator with NIST, DIEHARD, and ENT test suite. Figure 3d shows the Tinkerbell map implemented on C++. They found they could reach a speed of 0.49 Mbits/sec, which was larger than Yang et al. [41]. They produced a thousand sequences of 1 million bits and their entropy could reach up to 7.999 bits/byte and \(\chi ^2\) could reach up to 240.50, which gets better by 73 % with time.
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Wu and his team designed true random number generators based on semiconductor super-lattice chaos and further implemented it [45]. They could achieve a maximum bit rate of 300 Mbps for generating random bits. Wu and his team implemented the proposed mechanism using FPGA.
Table 2 shows the comparison of multiple methods reported in the literature that use chaotic maps for the generation of pseudo-random numbers and/or using them for encryption. The comparison was done on the basis of the correlation coefficient and entropy. For a random number generator, the correlation coefficient between its subsections ideally should be zero but for all practical reasons, it should be as low as possible. There was a wide range of correlation coefficients reported in literature highest being 0.0857 by Tang et al. [49] and the lowest being 0.00006 by Stoyanov et al. [20]. Though there is a difference of three orders of magnitude between the highest reported correlation coefficient and the lowest one. It is impressive that none of the methods has even crossed 0.1 as a correlation coefficient (not even a 10 % matching). This proves that chaotic maps are highly random. Thus making any chaotic map-based pseudo-random number generator highly random in nature and extremely difficult to break through the security provided by them. The ideal value of entropy should be 8 bits per byte. Most of the works in literature have reported an entropy value very close to this ideal value. The highest value of entropy was reported was 7.999995 by Akhshani et al. [26] and the lowest was reported was 7.98 by Pan et al. [34]. This proves that chaotic maps result in systems that can generate an output with a lot of information packed into one byte. This makes chaotic map-based systems extremely difficult to predict, hence making them suitable to be used for applications like OTP generation and encryption. The Ikeda map (shown in Fig. 3a) used by Stoyanov et al. [20] to develop a pseudo-random number generator has proven to be able to have a decent entropy with a very low value of correlation coefficient simultaneously.
There were different bifurcation maps, but mainly seen as a map that explodes or a map that remains flat. The exploding maps typically take larger fluctuating jumps at higher values of r. Whereas flat-type bifurcation diagrams can be used at very low values of r as well. Ideally, r should be chosen greater than 1.5 and within the range of 40, but this choice can be further user-defined as per the application. The hardware-based implementation of random-number generators totally depends on the number of bits the Arithmetic Logic Unit (ALU) of the microcontroller can handle. The speed of the random-number generator is controlled by the clock frequency of the controller. We found that at least 100 registers are used when the chaotic maps are implemented on the FPGA. The maximum operating frequency of an FPGA could reach 100 MHz and there is a well-defined research gap to reach at least a few GigaHertz frequencies of random number generation. Due to the complex mathematical calculation that a typical chaotic map has, the execution time of the entire sequence could not cross 0.2 seconds and we need pseudo-random number generators that can produce its entire file in a few milliseconds. Hence, chaotic map-based pseudo-random number generators are not suited for real-time applications but still can be used in applications like OTP generation and image encryption. Throughout the survey, we found that all the researchers achieved a correlation coefficient that was very close to the ideal value of zero and the entropy reported by maximum researchers crossed 7.99, whose ideal value is 8 bits per byte. We hope that people will keep on implementing the chaotic maps for different schemes in random generation and may find extremely secure systems built on it. This will help in the complete prohibition of electronic fraud. 2ff7e9595c
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