In this paper, we present a new attack on RSA when the public exponent is short, for instance 3 or 216+1, and when the classical exponent randomization is used. This attack works even if blinding is used on the messages. From a Simple Power Analysis (SPA) we study the problem of recovering the RSA private key when non consecutive bits of it leak from the implementation. We also show that such information can be gained from sliding window implementations not protected against SPA.