In recent years, cognitive Internet of Things (CIoT) has received considerable attention because it can extract valuable information from various Internet of Things (IoT) devices. In CIoT, truth discovery plays an important role in identifying truthful values from large scale data to help CIoT provide deeper insights and value from collected information. However, the privacy concerns of IoT devices pose a major challenge in designing truth discovery approaches. Although existing schemes of truth discovery can be executed with strong privacy guarantees, they are not efficient or cannot be applied in real-life CIoT applications. This article proposes a novel framework for lightweight and privacy-preserving truth discovery called LPTD-I, which is implemented by incorporating fog and cloud platforms, and adopting the homomorphic Paillier encryption and one-way hash chain techniques. This scheme not only protects devices’ privacy, but also achieves high efficiency. Moreover, we introduce a fault tolerant (LPTD-II) framework which can effectively overcome malfunctioning CIoT devices. Detailed security analysis indicates the proposed schemes are secure under a comprehensively designed threat model. Experimental simulations are also carried out to demonstrate the efficiency of the proposed schemes. •LPTD-I is proposed to defend against false data injection attacks.•An enhanced scheme, named LPTD-II, is presented to achieve fault tolerance.•Security analysis indicates the proposed schemes are secure.•Experimental simulations demonstrate the efficiency of the proposed schemes.