## 3 Proposed Chaos-Based Stream Ciphers Based on Secure PRNGs of Chaotic Sequences
### Security Analysis and Hardware Metrics of the Proposed PRNGs-CS  
#### 3.3.1 PRNGs-CS Resilience against Statistical Attacks  
A set of tests should be applied to assess the statistical cryptographic characteristic of the generated sequences by the proposed PRNGs-CS. Each test checks for a particular feature, such as the correlation between generated sequences or their uniformity, and the composite results of these tests give an insight into the level of randomness of the generated sequences. The statistical properties of the generated sequences are closely linked to the pseudo-chaotic behavior of these sequences. The National Institute of Standards and Technology (NIST) test, among other tests (TestU01, DieHARD), serves as a reference for quantifying and comparing the statistical effects of pseudo-random sequences. **Phase space test**  
We draw in Figure 9a-c the phase space or mapping of sequences X1, X2, and X3 generated by the suggested LSP-PRNG, LST-PRNG, and LSPT-PRNG, respectively, and created by 3,125,000 samples. In Figure 9d-f, we present the mapping of 1000 samples (a zoom) selected randomly from X1, X2, and X3, respectively. Figure 8: LSPT-PRNG design architecture proposed. Already, from Figure 9d-f, The area seems chaotic, indicating a lack of association between nearby sample results. **Histogram and Chi-square tests**  
A fundamental characteristic of a secure PRNG-CS is that the generated sequences are equally distributed. Histograms of the produced sequences X1, X2, and X3 are shown in Figure 10. Their uniformities are visually observed. Figure 10: Histograms of the produced sequences X1, X2, and X3 by LSP-PRNG, LST-PRNG, and LSPT-PRNG respectively. (**a**) The histogram of the produced sequence X1. (**b**) The histogram of the produced sequence X2. (**c**) The histogram of the produced sequence X3. Figure 9: Phase space and zoom of the produced sequences X1, X2, and X3 by LSP-PRNG, LST-PRNG, and LSPT-PRNG, respectively. (**a**) Mapping of sequence X1. (**b**) Mapping of sequence X2. (**c**) Mapping of sequence X3. (**d**) Zoom on the mapping of X1. (**e**) Zoom on the mapping of X2. (**f**) Zoom on the mapping of X3. The chi-square test, as defined by Equation (6), should be used to check visual uniformity results. Table 4 shows the experimental and theoretical chi-square tests for sequences X1, X2, and X3 generated by LSP-PRNG, LST-PRNG, and LSPT-PRNG, respectively. The experimental chi-square test values are lower than the theoretical values for all proposed PRNGs-CS, confirming the uniformity of the histograms. **NIST test**  
A further important effect of a secure PRNG-CS is that the generated sequences must pass the NIST statistical tests (previously defined in Section 2). The results obtained show that the sequences generated by the proposed PRNGs-CS pass all 15 statistical tests, as shown in Figure 11 and Table 5.  
**Key sensitivity analysis (using Hamming Distance)**  
Key sensitivity is a critical property for PRNGs. Of course, a slight modification in the secret key should result in a significant difference in the output sequence. To prove this \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Chi-Square Test Value** & **LSP-PRNG** & **LST-PRNG** & **LSPT-PRNG** \\ \hline \(\chi^{2}_{gh}\) & 1073.6426 & 1073.6426 & 1073.6426 \\ \(\chi^{2}_{exp}\) & 941.5878 & 896.3603 & 915.5385 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Chi-Square test. Figure 11: NIST test results of the proposed PRNGs-CS. (**a**) NIST test for LSP-PRNG. (**b**) NIST test for LST-PRNG. (**c**) NIST test for LSPT-PRNG.