In this paper we present new arithmetical and algebraic results following the work of Babindamana and al. on hyperbolas and describe in the new results an approach to attacking a RSA-type modulus based on continued fractions, independent and not bounded by the size of the private key $d$ nor the public exponent $e$ compared to Wiener's attack. When successful, this attack is bounded by $\displaystyle\mathcal{O}\left( b\log{\alpha_{j4}}\log{(\alpha_{i3}+\alpha_{j3})}\right)$ with $b=10^{y}$, $\alpha_{i3}+\alpha_{j3}$ a non trivial factor of $n$ and $\alpha_{j4}$ such that $(n+1)/(n-1)=\alpha_{i4}/\alpha_{j4}$. The primary goal of this attack is to find a point $\displaystyle X_{\alpha}=\left(-\alpha_{3}, \ \alpha_{3}+1 \right) \in \mathbb{Z}^{2}_{\star}$ that satisfies $\displaystyle\left\langle X_{\alpha_{3}}, \ P_{3} \right\rangle =0$ from a convergent of $\displaystyle\frac{\alpha_{i4}}{\alpha_{j4}}+\delta$, with $P_{3}\in \mathcal{B}_{n}(x, y)_{\mid_{x\geq 4n}}$. We finally present some experimental examples. We believe these results constitute a new direction in RSA Cryptanalysis using continued fractions independently of parameters $e$ and $d$.

翻译：本文在Babindamana等人关于双曲线研究的基础上，提出了新的算术与代数结果，并描述了一种基于连分数攻击RSA型模数的方法。与Wiener攻击相比，该方法独立于且不受私钥$d$或公钥指数$e$大小的限制。当攻击成功时，其复杂度为$\displaystyle\mathcal{O}\left( b\log{\alpha_{j4}}\log{(\alpha_{i3}+\alpha_{j3})}\right)$，其中$b=10^{y}$，$\alpha_{i3}+\alpha_{j3}$为$n$的非平凡因子，而$\alpha_{j4}$满足$(n+1)/(n-1)=\alpha_{i4}/\alpha_{j4}$。该攻击的主要目标是从$\displaystyle\frac{\alpha_{i4}}{\alpha_{j4}}+\delta$的渐近分数中寻找满足$\displaystyle\left\langle X_{\alpha_{3}}, \ P_{3} \right\rangle =0$的点$\displaystyle X_{\alpha}=\left(-\alpha_{3}, \ \alpha_{3}+1 \right) \in \mathbb{Z}^{2}_{\star}$，其中$P_{3}\in \mathcal{B}_{n}(x, y)_{\mid_{x\geq 4n}}$。我们最后给出了一些实验示例。我们相信这些结果为独立于参数$e$和$d$的基于连分数的RSA密码分析提供了新的研究方向。