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 $\displaystyle\mathcal{O}\left( b\log{α_{j4}}\log{(α_{i3}+α_{j3})}\right)$ with $b=10^{y}$, $α_{i3}+α_{j3}$ a non trivial factor of $n$ and $α_{j4}$ such that $(n+1)/(n-1)=α_{i4}/α_{j4}$. The primary goal of this attack is to find a point $\displaystyle X_α=\left(-α_{3}, \ α_{3}+1 \right) \in \mathbb{Z}^{2}_{\star}$ that satisfies $\displaystyle\left\langle X_{α_{3}}, \ P_{3} \right\rangle =0$ from a convergent of $\displaystyle\frac{α_{i4}}{α_{j4}}+δ$, 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 $d$.