Real-time price signals and power generation levels (disaggregated or aggregated) are commonly made available to the public by Independent System Operators (ISOs) to promote efficiency and transparency. However, they may inadvertently reveal crucial private information about the power grid, such as the cost functions of generators. Adversaries can exploit these vulnerabilities for strategic bidding, potentially leading to financial losses for power market participants and consumers. In this paper, we prove the existence of a closed-form solution for recovering coefficients in cost functions when LMPs and disaggregated power generation data are available. Additionally, we establish the convergence conditions for inference the quadratic coefficients of cost functions when LMPs and aggregated generation data are given. Our theoretical analysis provides the conditions under which the algorithm is guaranteed to converge, and our experiments demonstrate the efficacy of this method on IEEE benchmark systems, including 14-bus and 30-bus and 118-bus systems.