Implementing Wiener's Extensions in the Range $N^\frac{1}{4}\leq d <N^{\frac{3}{4}-\beta}$ and $N^{\frac{1}{4}}\leq d<N^{(\frac{1-\gamma}{2})}$, $\gamma\leq\frac{1}{2}$ with Lattice Reduction

Abstract

In this paper, Wiener Attack extensions on RSA are implemented with approximation via lattice reduction. The continued fraction based arguments of Wiener Attack extensions in the range $N^\frac{1}{4}\leq d <N^{\frac{3}{4}-\beta}$, $p-q=N^{\beta}$ and $N^{\frac{1}{4}}\leq d<N^{(\frac{1-\gamma}{2})}$, $|\rho q - p|\leq \frac{N^{\gamma}}{16}$, $1 \leq \rho \leq 2$, $\gamma\leq\frac{1}{2}$, are implemented with the Lattice based arguments and the LLL algorithm is used for reducing a basis of a lattice.