We discuss how to recover the private key for DSA style signature schemes if partial information about the ephemeral keys is revealed. The partial information we examine is of a second order nature that allows the attacker to know whether certain bits of the ephemeral key are equal, without actually knowing their values. Therefore, we extend the work of Howgrave-Graham, Smart, Nguyen and Shparlinski who, in contrast, examine the case where the attacker knows the actual value of such bits. We also discuss how such partial information leakage could occur in a real life scenario. Indeed, the type of leakage envisaged by our attack would appear to be feasible than that considered in the prior work.