# New Semi-Prime Factorization and Application in Large RSA Key Attacks
## 5 Case Study Examples Applied to RSA Key Factorization
\begin{table}
\begin{tabular}{c c} \hline \hline \multirow{2}{*}{\(c_{1}c_{2}\) RSA768} & 123018668453011775513049495838496272077285356995533479219732245215 \\  & 17264005072636575187452021997864693899564749427406384592519255732 \\  & 63034537315482685079170261221429134616704292143160222124047927473 \\  & 7794080665351419597459856902143413 \\ \hline \multirow{6}{*}{\(c_{1}c_{2})^{2}\)} & 151335927879520346290803999457322598363508796074958431058717144380 \\  & 24587283531274572135724637572332720319718269519140130366434717995 \\  & 557448975805325859290191578969849347584921949069093738365207236 \\  & 511522080338199512854710820317535056362120218189196094883472059867 \\  & 67027371128840419120676100828310793615855810542275588818703810974 \\  & 813042919826944946811902964522404849739464632596646341875606194756 \\  & 2985467999013006479462484696511372504488571635780855919793619288569 \\ \hline \multirow{6}{*}{\(\beta\)} & 122282464059209245992174346897658189706307595059668582419459721623 \\  & 5752683964481520057217289973858098738702956370826573062458727992772 \\  & 2157915357894291663323174826804895133562988515599077049551048 \\  & 7891461504154220362780972595368188 \\ \hline \multirow{6}{*}{\(\sqrt{\frac{\alpha}{\left(c_{1}c_{2}\right)^{2}-\beta^{2}}}\)} & 1343844379223531027642685110944406395128150457221349204815473265992 \\  & 860466317856575453545416265884545802321235838004890967990322730328 \\ \cline{1-1}  & 63977802038544981826008347677792634734515510595044348205814798 \\ \cline{1-1}  & 59356430564624411389559844289235 \\ \hline \multirow{2}{*}{\(c_{1}gcd(\alpha,\beta)\)} & 33478071698956989786044169848216908177047949837137685689124313889 \\ \cline{1-1}  & 82883793878002287614711652531743087737814467999489 \\ \hline \multirow{2}{*}{\(c_{2}\frac{\text{BSA768}}{c_{1}}\)} & 36746043666799590428244633799627952632795186143430876426760322838 \\ \cline{1-1}  & 1573966651127923337341714396810270092798736308917 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Application of the proposed semi-prime factorization method for RSA-768 attack. * [6] [PERSON]; [PERSON] Polynomial-time solution of prime factorization and NP-complete problems with digital memcomputing machines. _Chaos Interdiscip. J. Nonlinear Sci._**2017**, _27_, 023107. [CrossRef] [PubMed]
* [7] [PERSON]; [PERSON] Puzzle--Solving the n-Fractions Puzzle as a Constraint Programming Problem. _INFORMS Trans. Educ._**2018**, _19_, 48-55. [CrossRef]
* [8] [PERSON] _SSL and TLS: Designing and Building Secure Systems_; [PERSON]: London, UK, 2001. * [9] [PERSON] _Applied Cryptography_, 2 nd ed. ; John Wiley & Sons, Inc.: New York, NY, USA, 1996. * [10] [PERSON]; [PERSON]; [PERSON] A method for obtaining digital signatures and public-key cryptosystems. _Commun. ACM_**1978**, _21_, 120-126. [CrossRef]
* [11] [PERSON]; [PERSON]; [PERSON]; [PERSON] Dual RSA and Its Security Analysis. _IEEE Trans. Inf. Theory_**2007**, _53_, 2922-2933. * [12] [PERSON] Turning Euler's Factoring Method into a Factoring Algorithm. _Bull. Lond. Math. Soc._**1996**, _28_, 351-355. [CrossRef]
* [13] [PERSON] A One-Sentence Proof That Every Prime p \(\equiv\) 1 (mod 4) Is a Sum of Two Squares. _Am. Math. Mon._**1990**, _97_, 144. * [14] [PERSON] _The Sum of Two Squares_; Cornell University Press: New York, NY, USA, 2013. * [15] [PERSON] Pythagorean Triples before and after [PERSON]. _Computation_**2020**, \(8\), 62. [CrossRef]
* [16] [PERSON] Twenty years of attacks on the RSA cryptosystem. _Not. Am. Math. Soc. (AMS)_**1999**, _46_, 203-213. * [17] [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Factoring as a Service. In _Financial Cryptography and Data Security. FC 2016. Lecture Notes in Computer Science_; [PERSON], [PERSON], [PERSON]; Springer: Berlin, Germany, 2017; Volume 9603. * [18] [PERSON]; [PERSON]; [PERSON]; [PERSON] Analysis of the HTTPS certificate ecosystem. In Proceedings of the 13 th Internet Measurement Conference, Barcelona, Spain, 23-25 October 2013. * [19] [PERSON] Cryptanalysis of short RSA secret exponents. _IEEE Trans. Inf. Theory_**1990**, _160_, 553-558.