# New Semi-Prime Factorization and Application in Large RSA Key Attacks
## 5 Case Study Examples Applied to RSA Key Factorization  
In this section, we illustrate the application of our proposed semi-prime factorization method using case study examples. An application area of particular interest in considering specific variants of RSA is how both small and large encryption keys perform within our proposed factorization approach. With the evolution of the Internet of things (IoT), the emergence of lightweight cryptography is on the rise. However, due to the relatively low computational power of personal devices, malicious attacks are recently targeting IoT networks [62, 63, 64]. Since the security of cryptographic operations in both small keys as well as large keys depends upon whether the semi-prime factorization can be solved efficiently, we consider one application example for each of the two scenarios. **Application Example 1. ** Let us consider Case Example 2 with N = 377. The 45 Pythagorean quadruples are:  
(12,81,368,377), (12,108,361,377), (12,156,343,377), (12,224,303,377), (15,252,280,377), (17,144,348,377), (17,192,324,377), (24,143,348,377), (24,177,332,377), (28,72,369,377), (28,252,279,377), (39,72,368,377), (39,208,312,377), (44,207,312,377), (44,228,297,377), (64,252,273,377), (72,199,312,377), (72,252,271,377), (81,108,**352**,377), (84,132,343,377), (84,208,303,377), (87,116, **348**,377), (87,156,332,377), (87,172,324,377), (89,192,312,377), (100,105,**348**,377), (100,240,**273**, 377), (105,252,**260**,377), (108,207,296,377), (108,233,276,377), (116,177,312,377), (116,192,303, 377), (132,224,**273**,377), (143,156,312,377), (143,228,264,377), (144,172,303,377), (**145**,240,252, 377), (152,207,276,377), (156,172,297,377), (156,208,**273**,377), (156,233,252,377), (172,207,264, 377), (177,180,280,377), (180,215,252,377), (192,208,249,377)  
\(\beta\in\{12,\) 15, 17, 24, 28, 39, 44, 64, 72, 81, 84, 87, 89, 100, 105, 108, 116, 132, 143, 144, **145**, **152**, 156, 172, 177, 180, 192, 207, 208, 215, 224, 228, 233, 240, 249, 252, **260**, 264, 271, **273**, 276, 279, 280, 296, 297, 303, 312, 324, 332, 343, **348**, **352**, 361, 368, 369\}  
From Lemma 4 and Equation (20) we have:  
\[\beta\in(z_{1},z_{2},z_{3})\
i(\alpha,\beta,N)\ :\alpha^{2}=N^{2}-\beta^{2},\ \alpha\
eq\beta\]  
The four Pythagorean triples are:  
(**135**,352**,377), (145,348,377), (152,345,377), (260,273,377)  
Note that:  
\[\exists\alpha:\{\alpha\}\
subseteq\{\beta\}\ :\{135,\ 345\}\]  
\(\
exists\{\beta\}\) with \(\gcd(\alpha,\beta)=1\) are not solutions to the semi-prime. However, solutions can be found as follows [1,14]:  
\[\{\alpha\}\
subseteq\{\beta\}\ \Rightarrow\gcd(\alpha,\beta)=1,\]  
\[(135,352,377)\ \Rightarrow\gcd(135,352)=1,\]  
\[x_{1}=\sqrt{\left(\frac{377-135}{2}\right)}=11,\ x_{2}=\sqrt{377-11^{2}}=16\]  
\[377=11^{2}+16^{2}\]  
\[(152,345,377)\ \Rightarrow\gcd(152,345)=1\]  
\[x_{3}=\sqrt{\left(\frac{377-345}{2}\right)}=4,\ x_{4}=\sqrt{377-4^{2}}=19\]  
\[377=4^{2}+19^{2}\]  
\[o=19-11=8,\ e=16-4=12,\ g=\gcd(8,12)=4\]\[\exists\alpha:\{\alpha\}\
subseteq\{\beta\}\ :\{135,\ 345\}\]0  
\[\exists\alpha:\{\alpha\}\
subseteq\{\beta\}\ :\{135,\ 345\}\]1  
From Lemma 3 and Equation (15) we have:  
\[\exists\alpha:\{\alpha\}\
subseteq\{\beta\}\ :\{135,\ 345\}\]2  
\[\exists\alpha:\{\alpha\}\
subseteq\{\beta\}\ :\{135,\ 345\}\]3  
\[\exists\alpha:\{\alpha\}\
subseteq\{\beta\}\ :\{135,\ 345\}\]4  
\[\exists\alpha:\{\alpha\}\
subseteq\{\beta\}\ :\{135,\ 345\}\]5  
\[\exists\alpha:\{\alpha\}\
subseteq\{\beta\}\ :\{135,\ 345\}\]6  
From Lemma 2 we have:  
The Pythagorean triples with prime factors from Equation (14) are given as follows:  
\[\exists\alpha:\{\alpha\}\
subseteq\{\beta\}\ :\{135,\ 345\}\]7  
\[\exists\alpha:\{\alpha\}\
subseteq\{\beta\}\ :\{135,\ 345\}\]8  
\[\exists\alpha:\{\alpha\}\
subseteq\{\beta\}\ :\{135,\ 345\}\]9  
\[\{\alpha\}\
subseteq\{\beta\}\ \Rightarrow\gcd(\alpha,\beta)=1,\]0  
Figure 1 provides a pictorial representation of Pythagorean triples with prime factors for the case example with \(N=377\). **Application Example 2.