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In this paper, based on a Linear Congruential Generator (LCG), we propose a new block cipher that is suitable for constructing a lightweight secure protocol for resource-constrained wireless sensor networks. From the cryptanalysis point of view, our building block is considered secure if the attacker cannot obtain the pseudo-random numbers generated by the LCG. The Plumstead's inference algorithm for a LCG with unknown parameters demonstrates that it is impossible to significantly enhance the security of the system simply by increasing the size of the modulus. Therefore, we are motivated to embed the generated pseudo-random numbers with sensor data messages in order to provide security. Specifically, the security of our proposed cipher is achieved by adding random noise and random permutations to the original data messages. We also adopt the Hull and Dobell's algorithm to select proper parameters used in the LCG. The analysis of our cipher indicates that it can satisfy the security requirements of wireless sensor networks. We further demonstrate that secure protocols based on our proposed cipher satisfy the baseline security requirements: data confidentiality, authenticity, and integrity with low overhead. Performance analysis demonstrates that our proposed block cipher is more lightweight than RC5, a commonly used cipher in wireless sensor networks, in terms of the number of basic operations.