The Internet of Things (IoT) features a mechanism that extends connectivity to diverse computing devices, such as smart phones, commodity sensors, and appliances. Due to the huge quantity of data generated by the IoT devices, they are likely to be stored and managed by the cloud these days. However, because of the privacy concern about the sensitive data, encryption techniques are typically adopted by the cloud. In order to enable searching over encrypted data for multiple data senders in the cloud, public key encryption with keyword search (PEKS) has been proposed as one variant of searchable encryption (SE). Unfortunately, existing PEKS schemes are vulnerable to adaptive file-injection attack due to the lack of forward privacy. In this paper, we propose a forward secure PEKS scheme based on hierarchical identity-based encryption for cloud-assisted IoT environments. While the existing schemes incur to the data receiver a storage overhead that increases linearly with the number of data senders, our scheme incurs only a constant cost. The experimental analysis with Amazon EC2 and Raspberri Pi shows that our scheme is two to five times more efficient than the previous schemes, which makes our scheme more suitable for multiple data senders in the cloud-assisted IoT environments.