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 δ ), where we are mostly interested in the range 0.3 ≤ δ ≤ 0.5. We use both the upper and lower bounds on φ(N) and then try to find out what are the cases when is a convergent in the CF expression of . First we show that the RSA keys are weak when d∈=∈N δ and , where 2q∈-∈p∈= ∈N γ and τ is a small value based on certain parameters. This presents additional results over the work of de Weger (AAECC 2002). Further we show that, the RSA keys are weak when and e is for . Using similar idea we also present new results over the work of Blömer and May (PKC 2004). © 2008 Springer-Verlag Berlin Heidelberg.