**Revisiting Wiener's Attack -- New Weak Keys in RSA**

*Subhamoy Maitra and Santanu Sarkar*

**Abstract: **In this paper we revisit Wiener's method (IEEE-IT 1990) of continued fraction (CF) to find new weaknesses in RSA. We consider RSA with $N = pq$, $q < p < 2q$, public encryption exponent $e$ and private decryption exponent $d$. Our motivation is to find out when RSA is insecure given $d$ is $O(N^\delta)$, where we are mostly interested in the range $0.3 \leq \delta \leq 0.5$. Given $\rho$ $(1 \leq \rho \leq 2)$ is known to the attacker, we show that the RSA keys are weak when $d = N^\delta$ and $\delta < \frac{1}{2} - \frac{\gamma}{2}$, where $|\rho q - p| \leq \frac{N^\gamma}{16}$. This presents additional results over the work of de Weger (AAECC 2002). We also discuss how the lattice based idea of Boneh-Durfee (IEEE-IT 2000) works better to find weak keys beyond the bound $\delta < \frac{1}{2} - \frac{\gamma}{2}$. Further we show that, the RSA keys are weak when $d < \frac{1}{2} N^\delta$ and $e$ is $O(N^{\frac{3}{2}-2\delta})$ for $\delta \leq \frac{1}{2}$. Using similar techniques we also present new results over the work of Bl{\"o}mer and May (PKC 2004).

**Category / Keywords: **public-key cryptography / Cryptanalysis, RSA, Factorization, Weak Keys.

**Publication Info: **ISC 2008, 11th Information Security Conference, September 15-18, 2008, Taipei, Taiwan, to be published in Lecture Notes in Computer Science, Springer, 2008.

**Date: **received 19 May 2008, last revised 19 Feb 2009

**Contact author: **subho at isical ac in

**Available format(s): **PDF | BibTeX Citation

**Note: **Substantial Revision.

**Version: **20090220:065931 (All versions of this report)

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