In this paper we present new arithmetical and algebraic results following the work of Babindamana and al. on hyperbolas and describe from 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 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.

翻译：本文在Babindamana等人关于双曲线研究的基础上，提出新的算术与代数结果，并基于这些结果描述了一种针对RSA型模数的攻击方法。该方法基于连分数，与Wiener攻击不同，其有效性不受私钥$d$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，$\alpha_{j4}$满足$(n+1)/(n-1)=\alpha_{i4}/\alpha_{j4}$。该攻击的核心目标是从$\displaystyle\frac{\alpha_{i4}}{\alpha_{j4}}+\delta$的渐近分数中找到一个点$\displaystyle X_{\alpha}=\left(-\alpha_{3}, \ \alpha_{3}+1 \right) \in \mathbb{Z}^{2}_{\star}$，使其满足$\displaystyle\left\langle X_{\alpha_{3}}, \ P_{3} \right\rangle =0$，其中$P_{3}\in \mathcal{B}_{n}(x, y)_{\mid_{x\geq 4n}}$。最后我们给出若干实验实例。我们认为这些结果构成了利用连分数进行RSA密码分析的新方向。