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Database-driven Dynamic Spectrum Sharing (DSS) is the de-facto technical paradigm adopted by Federal Communications Commission (FCC) for increasing spectrum efficiency. In such a system, a geo-location database administrator (DBA) maintains spectrum availability information over its service region whereby to determines whether a secondary user can access a licensed spectrum band at his desired location and time. To maintain spectrum availability in its service region, it is desirable for the DBA to periodically collect spectrum measurements whereby to construct and maintain a Radio Environment Map (REM), where the received signal strength at every location of interest is either directly measured or estimated via proper statistical spatial interpolation techniques. Crowdsourcing-based spectrum sensing is a promising approach for periodically collecting spectrum measurements over a large geographic area, which is, unfortunately, vulnerable to false spectrum measurements. How to construct an accurate REM in the presence of false measurements remains an open challenge. This paper introduces SecREM, a novel scheme for securely constructing a REM in the presence of false spectrum measurements. SecREM relies on a small number of trusted spectrum measurements whereby to evaluate the trustworthiness of the measurements from mobile users and gradually incorporate the most trustworthy ones to construct an accurate REM. Extensive simulation studies based on a real spectrum measurement dataset confirm the efficacy and efficiency of SecREM.