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Public intelligent services enabled by machine learning algorithms are vulnerable to model extraction attacks that can steal confidential information of the learning models through public queries. Differential privacy (DP) has been considered a promising technique to mitigate this attack. However, we find that the vulnerability persists when regression models are being protected by current DP solutions. We show that the adversary can launch a query-flooding parameter duplication (QPD) attack to infer the model information by repeated queries. To defend against the QPD attack on logistic and linear regression models, we propose a novel High-Dimensional Gaussian (HDG) mechanism to prevent unauthorized information disclosure without interrupting the intended services. In contrast to prior work, the proposed HDG mechanism will dynamically generate the privacy budget and random noise for different queries and their results to enhance the obfuscation. Besides, for the first time, HDG enables an optimal privacy budget allocation that automatically determines the minimum amount of noise to be added per user-desired privacy level on each dimension. We comprehensively evaluate the performance of HDG using real-world datasets and shows that HDG effectively mitigates the QPD attack while satisfying the privacy requirements. We also prepare to open-source the relevant codes to the community for further research.