## 6 Applications of Chaos to Encryption
### Applications of Chaos to Random Number Generation  
With the rapid development of communication technology, it is worth noting that random numbers are used in many cryptographic protocols, including key management, identity authentication and image encryption. Since randomness and unpredictability are required by encryption, it is clear that chaos has good features in this regard, by virtue of the sensitive dependence on initial conditions, periodicity and reproduction. Some interesting manuscripts have been recently published on the topic [27; 28; 29; 30; 31; 32; 33; 34; 35]. For example, in [27] the dynamics of some discrete-time chaotic systems in the digital (i.e., finite-precision) domain is studied. Differently from classical approaches treating a digital chaotic map as a black box, in [27] the dynamical properties of the logistic map and the tent map are explored with fixed-point arithmetic from a new perspective. Namely, the innovative approach is based on state-mapping networks, where each value in the digital domain is treated as a node, whereas the mapping between pairs of nodes is treated as a directed edge [27]. Scale-free properties of the logistic map under the state-mapping network approach are clearly highlighted. Moreover, extension of the technique to floating-point arithmetic is studied in detail [27]. Finally, the authors of reference [27] highlight that to understand the state-mapping network structure of chaotic discrete-time systems in digital computers, it may help to take into account unwelcome degenerations of chaos in finite-precision domains. This may help to enhance the randomness of pseudorandom number generators based on chaotic maps [27]. In [28], it is pointed out that low-power devices used in Internet-of-things networks currently lack security owing to the high power consumption of random number generators. Consequently, the authors of reference [28] present a low-power true random number generator that exploits a four-dimensional hyperchaotic system characterized by hidden attractors. A customized circuit is realized using 130 nm CMOS technology, with the aim to integrate it into low-power devices. Moreover, chaotic data post-processing via SHIFT-XOR is exploited to produce random bit outputs [28]. The consumption of the hyperchaotic circuit reaches the maximum value of 980 uW to generate the chaotic signals, whereas the dissipation of the static current is about 623 uA. Finally, the authors of reference [28] highlight that the approach guarantees ready-to-use random bit sequences that passed the randomness test NIST SP800-22. A novel chaotic system, which exploits a recently proposed signum thermostat, is presented in [29]. The implemented circuit produces a signal whose probability distribution function is accurately Gaussian. Data from the implemented circuit are analysed. The experimental results prove to be in agreement with the theoretical analysis, indicating that the implemented chaotic circuit is effective in producing accurate Gaussian random numbers [29]. In [30], a simple method for designing pseudo-random bit generators is presented. The approach enables multiple bits per iteration to be generated from the decimal part of a chaotic map [30]. The objective is achieved by removing the decimal part of the state in each iteration and by comparing each digit with respect to a threshold value. By using such a technique, more than one bit can be generated in each iteration. This represents an advantage over most of the chaos-based generators available in the literature, since they can produce only one bit [30]. The final bitstream passes all NIST tests. Moreover,simulation results show that for each iteration about 8 digits can be extracted for most of the chaotic maps, indicating the effectiveness of the conceived approach in designing innovative pseudo-random bit generators [30]. By using fuzzy triangular numbers, in [31] a modification of the well-known logistic map is proposed.