## 2 Study of Used Discrete Chaotic Maps: Statistical Test and Hardware Metrics
### Discrete Equations of the Chaotic Maps and the 32-Bit LFSR Used  
We give below the discrete equations of the chaotic maps, and the 32-bit LFSR used to design the proposed chaotic systems:  
**Discrete Logistic map**  
The logistic map, created by [PERSON] in 1845, was initially designed as a model for population growth [35]. Due to the recurrence equation's simplicity, [PERSON] and [PERSON] in 1947 used the logistic map as a pseudo-random number generator [36]. Since then, it has been widely used in cryptographic applications. When the control parameter value is set to 4, the discrete Logistic map equation is as follows:  
\[XL(n)=\left\{\begin{array}{cc}\left\lfloor\frac{XL(n-1)\left[2^{N}-XL(n-1) \right]}{2^{N-2}}\right\rfloor&if\;XL(n-1)\
eq\left[3\times 2^{N-2}-1,\,2^{N-1} \right]\\ \\ 3\times 2^{N-2}-1&if\;XL(n-1)=3\times 2^{N-2}\\ \\ 2^{N}-1&if\;XL(n-1)=2^{N-1}\end{array}\right. \tag{1}\]  
where \(XL(n)\) is an integer that falls within the range \([1,2^{N}-1]\) with \(N=32\) as the precision used and \(\left\lfloor Z\right\rfloor\) (_Floor function_) returns the highest integer that is less than or equal to \(Z\).  
**Discrete Skew-Tent map**  
The Skew-Tent map is a piecewise linear map, the equation of the discretized Skew-Tent map is defined by:  
\[XS(n)=\left\{\begin{array}{cc}\left\lfloor\frac{2^{N}\timesXS(n-1)}{P_{s}} \right\rfloor&if\;0<XS(n-1)<P_{s}\\ \\ \left\lfloor 2^{N}\times\frac{\left[2^{N}-XS(n-1)\right]}{2^{N}-P_{s}}\right\rfloor &if\;P_{s}<XS(n-1)<2^{N}\\ \\ 2^{N}-1&otherwise\end{array}\right. \tag{2}\]  
where \(XS(n)\) is an integer that falls within the range \([1,2^{N}-1]\) and \(P_{s}\) is the Skew-Tent map's control parameter, with a value in the range \([1,2^{N}-1]\).  
**Discrete PWLCM map**  
Another piecewise linear chaotic map is the Piecewise Linear Chaotic Maps (PWLCM); due to its good cryptographic characteristics, compared to the other chaotic maps studiedin this section, the PWLCM map is commonly employed in data encryption. The discrete PWLCM map's equation is as follows:  
\[XP(n)=\left\{\begin{array}{ccc}\left\lfloor\frac{2^{N}\times XP(n-1)}{P_{p}} \right\rfloor&&if\,0<XP(n-1)<P_{p}\\ \\ \left\lfloor 2^{N}\times\frac{\left\lfloor XP(n-1)-P_{p}\right\rfloor}{2^{N-1} -P_{p}}\right\rfloor&&if\,P_{p}<XP(n-1)<2^{N-1}\\ \\ \left\lfloor 2^{N}\times\frac{\left\lfloor 2^{N}-XP(n-1)-P_{p}\right\rfloor}{2^{N-1} -P_{p}}\right\rfloor&&if\,2^{N-1}<XP(n-1)<2^{N}-P_{p}\\ \\ \left\lfloor 2^{N}\times\frac{\left\lfloor 2^{N}-XP(n-1)\right\rfloor}{P_{p}} \right\rfloor&&if\,2^{N}-P_{p}<XP(n-1)<2^{N}\\ \\ 2^{N}-1&&otherwise\end{array}\right. \tag{3}\]  
where \(P_{p}\) is the PWLCM map's control parameter, with a value in the range \([1,2^{N-1}-1]\) and \(XP\) belongs to the same interval as \(XS\) and \(XL\).  
**Discrete 3D-Chebyshev map**  
The discrete 3D-Chebyshev (third order) map is given by:  
\[XT(n)=\left\lfloor 2^{(-2N+2)}\times\left(4\times\left(XT-2^{(N-1)}\right)^{3} -3\times 2^{(2N-2)}\times\left(XT-2^{(N-1)}\right)\right)+2^{(N-1)}\right\rfloor \tag{4}\]  
where \(1\leq XT(n)\leq 2^{N}-1\).  
**32-bit LFSR based on primitive polynomial with internal feedback (Galois implementation)**  
The 32-bit LFSR with maximum length feedback used in our work is defined by the following primitive polynomial:  
\[Q(x)=x^{32}+x^{22}+x^{2}+x+1 \tag{5}\]  
Its circuit diagram is shown in Figure 1. The sequences generated (random outputs) are periodic of period equal to \(2^{32}-1=4\),\(294\),\(967\),\(295\).