# Improved Cryptanalysis of Some RSA Variants
## 3 Main Results  
In this section, we introduce an efficient technique for breaking RSA variants characterized by the generalized key equation  
\[eu\equiv z\pmod{((p^{2}-1)(q^{2}-1))}.\]  
_The New Attack_  
**Theorem 3**. : _Let \((N,e)\) denote an RSA public key where \(N=pq\), \(q<p<2q\), and \(e=N^{\pi}\). Suppose that there are two parameters, \(|u|\leq N^{\delta}\) and \(|z|\leq N^{\gamma}\), such that \(eu\equiv z\pmod{\psi(N)}\), with \(\psi(N)=(p^{2}-1)(q^{2}-1)\). Assume also that \(p\) is approximated by \(\tilde{p}\) with an error bound of \(|p-\tilde{p}|\leq N^{\sigma}\), where \(0\leq\sigma\leq\frac{1}{2}\). If_  
\[2\sigma<\alpha<\frac{2}{\sigma}\quad\text{and}\quad\delta+\frac{\gamma}{2}<2- \sqrt{2\sigma\alpha},\]  
_then the factorization of \(N\) can be performed in polynomial time._  
Proof. : Let \(\tilde{p}\) be a close approximation of \(p\) such that \(|p-\tilde{p}|\leq N^{\sigma}\). By Lemma 2, the value \(\tilde{q}=\lfloor N/\tilde{p}\rfloor\) serves as an approximation for \(q\), satisfying the bounds  
\[|q-\tilde{q}|<N^{\sigma},\quad|p+q-\tilde{p}-\tilde{q}|<2N^{\sigma}.\]  
Let \(x\) be an integer satisfying \(eu-x\psi(N)=z\). This can be rewritten as  
\[-x\psi(N)\equiv z\pmod{e}.\]  
Write \(-\psi(N)=y^{2}+ay+b\), for  
\[a=2(\tilde{p}+\tilde{q}),\ b=(\tilde{p}+\tilde{q})^{2}-(N+1)^{2},\ y=p+q- \tilde{p}-\tilde{q}.\]  
As a consequence, the triplet  
\[\Big{(}x_{1}^{(0)},x_{2}^{(0)},x_{3}^{(0)}\Big{)}=(x,y,z),\]  
is a solution of the equation  
\[x_{1}(x_{2}^{2}+ax_{2}+b)\equiv x_{3}\pmod{e}.\]To compute the root \(\left(x_{1}^{(0)},x_{2}^{(0)},x_{3}^{(0)}\right)\), we employ an extended version of [PERSON]'s method. This involves analyzing the polynomial equation  
\[f(x_{1},x_{2},x_{3})\equiv 0\pmod{e},\]  
where  
\[f(x_{1},x_{2},x_{3})=x_{1}(x_{2}^{2}+ax_{2}+b)-x_{3}.\]  
Let \(x_{4}=x_{1}x_{2}^{2}-x_{3}\). Then, \(f(x_{1},x_{2},x_{3})=g(x_{1},x_{2},x_{4})\), where  
\[g(x_{1},x_{2},x_{4})=x_{1}(ax_{2}+b)+x_{4}.\]  
Building on [PERSON]'s method, we introduce an additional parameter \(\zeta>0\), which will be optimized, along with two integers \(\mu\) and \(w\), where \(0\leq w\leq\mu\). We then define the following list of polynomials:  
\[2\sigma<\alpha<\frac{2}{\sigma}\quad\text{and}\quad\delta+\frac{\gamma}{2}<2- \sqrt{2\sigma\alpha},\]0  
where  
\[2\sigma<\alpha<\frac{2}{\sigma}\quad\text{and}\quad\delta+\frac{\gamma}{2}<2- \sqrt{2\sigma\alpha},\]1 \[2\sigma<\alpha<\frac{2}{\sigma}\quad\text{and}\quad\delta+\frac{\gamma}{2}<2- \sqrt{2\sigma\alpha},\]2  
and, in the computations, the term \(x_{1}x_{2}^{2}\) is replaced by \(x_{4}+x_{3}\). Given that \(\left(x_{1}^{(0)},x_{2}^{(0)},x_{3}^{(0)}\right)\) is a root of the equation  
\[2\sigma<\alpha<\frac{2}{\sigma}\quad\text{and}\quad\delta+\frac{\gamma}{2}<2- \sqrt{2\sigma\alpha},\]3  
Then, by defining \(x_{4}^{(0)}=x_{1}^{(0)}\left(x_{2}^{(0)}\right)^{2}-x_{3}^{(0)}\), the triplet \(\left(x_{1}^{(0)},x_{2}^{(0)},x_{4}^{(0)}\right)\) satisfies the equation \(g(x_{1},x_{2},x_{4})\equiv 0\pmod{e}\). Furthermore, we observe that  
\[2\sigma<\alpha<\frac{2}{\sigma}\quad\text{and}\quad\delta+\frac{\gamma}{2}<2- \sqrt{2\sigma\alpha},\]4  
for all \((w,a,b)\in\mathcal{C}\cup\mathcal{D}\). In [PERSON]'s technique, it is important to establish the bounds \(X_{1}\), \(X_{2}\), \(X_{3}\), and \(X_{4}\) such that  
\[2\sigma<\alpha<\frac{2}{\sigma}\quad\text{and}\quad\delta+\frac{\gamma}{2}<2- \sqrt{2\sigma\alpha},\]5  
Using \(\psi(N)>\frac{N^{2}}{2}\) and assuming that \(|z|<e|u|\), we obtain the following estimate:  
\[2\sigma<\alpha<\frac{2}{\sigma}\quad\text{and}\quad\delta+\frac{\gamma}{2}<2- \sqrt{2\sigma\alpha},\]6  
To obtain a bound for \(x_{4}^{(0)}=x_{1}^{(0)}\left(x_{2}^{(0)}\right)^{2}-x_{3}^{(0)}\), we suppose  
\[2\sigma<\alpha<\frac{2}{\sigma}\quad\text{and}\quad\delta+\frac{\gamma}{2}<2- \sqrt{2\sigma\alpha},\]7  
so that \(\left|x_{4}^{(0)}\right|<2x_{1}^{(0)}\Big{(}x_{2}^{(0)}\Big{)}^{2}\).