This paper presents a novel discrete-time (DT) chaotic map-based random-number generator (RNG), namely the Siponi map, which is a modification of the Logistic map. The Logistic map is usually applied to cryptosystems, mainly for the purposes of generating random numbers. In addition to being easy to implement, it has a better security level than other nonlinear functions. However, it can only process positive real-number inputs. Our proposed map is a deterministic function that can process positive and negative real values. We explored the map comprehensively and investigated its characteristics and parameters. We calculated the optimum parameter values using empirical and theoretical mathematical models to yield the maximum randomness of a sequence of bits. The limit variation of the maximum parameter value was determined based on a practical information measure. Empirical verification was performed for the Siponi map to generate bit sequences unrelated to the previous bit with high entropy values, and we found the extractor function threshold value to be 0.5, while the parameter control was −2 or 2. Using our proposed map, a simple RNG without post-processing passed DieHard statistical tests and all the tests on the NIST SP 800-22. Finally, we have implemented a Siponi map-based RNG on the FPGA board and demonstrated that the sources used are LUT = 4086, DSP = 62, and register = 2206.

Substitution-box (s-box) is a basic component of block cipher which performs a substitution. Two powerful cryptanalysis techniques applied to block ciphers are linear cryptanalysis and differential cryptanalysis. The resistance against differential cryptanalysis can be achieved by eliminating high-probability differential trails. We should choose an s-box where the maximum difference propagation probability is as small as possible to eliminating high-probability differential trails. Nyberg proposed a method to construct the s-box by using the inverse mapping on a finite field then implements affine transformations on . In this study, we generate 47.104 s-box according to Nyberg. The experimental results showed that s-boxes have the maximum difference propagation probability with the same frequency.

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