One of the the most propitious enablers of the Smart Grid paradigm, and a rather low-hanging fruit, is Demand Response (DR) programs. DR is a set of courses of actions to reach and maintain the supply-demand equilibrium of the power grid from the side of the demand through price or incentive-based routines. The latter includes many programs with some bidding component. The general consensus among the experts is that the impact of the benefits is multiplied as DR program participation from the consumption side increases. However, there is an increasing awareness in public about privacy implications of many technological implementations including SG and DR deployments. In this paper, we are proposing a fully private bidding mechanism as part of a DR program to protect all the bidding details except for the winning price and winner at the end of the auction. To the best of our knowledge, ours is the first approach in the literature that accomplishes full privacy in a DR program without relying on a trusted third party. Preliminary simulation results show that the proposed mechanism presents a feasible solution for day-ahead and hour-ahead bidding mechanisms in the SG.