As network security monitoring grows more sophisticated, there is an increasing need for outsourcing such tasks to third-party analysts. However, organizations are usually reluctant to share their network traces due to privacy concerns over sensitive information, e.g., network and system configuration, which may potentially be exploited for attacks. In cases where data owners are convinced to share their network traces, the data are typically subjected to certain anonymization techniques, e.g., CryptoPAn, which replaces real IP addresses with prefix-preserving pseudonyms. However, most such techniques either are vulnerable to adversaries with prior knowledge about some network flows in the traces, or require heavy data sanitization or perturbation, both of which may result in a significant loss of data utility. In this paper, we aim to preserve both privacy and utility through shifting the trade-off from between privacy and utility to between privacy and computational cost. The key idea is for the analysts to generate and analyze multiple anonymized views of the original network traces; those views are designed to be sufficiently indistinguishable even to adversaries armed with prior knowledge, which preserves the privacy, whereas one of the views will yield true analysis results privately retrieved by the data owner, which preserves the utility. We present the general approach and instantiate it based on CryptoPAn. We formally analyze the privacy of our solution and experimentally evaluate it using real network traces provided by a major ISP. The results show that our approach can significantly reduce the level of information leakage (e.g., less than 1\% of the information leaked by CryptoPAn) with comparable utility.