We consider full-truthful Nash implementation, which requires that truth telling by each agent should be a Nash equilibrium of a direct revelation mechanism, and that the set of Nash equilibrium outcomes of the mechanism should coincide with the f-optimal outcome. We show that restricted monotonicity together with an auxiliary condition called boundedness is both necessary and sufficient for full-truthful Nash implementation. We also prove that full-truthful Nash implementation is equivalent to secure implementation(Saijo et al. (2005)). This gives us an alternative characterization of securely implementable social choice functions.