The Internet of Things (IoT) needs complex sensor networks. The increasing number of sensors and their networks underlines the importance of network security. Traditional digital signature schemes validate signatures from each node of networks, which need huge resources for storage and computation. To reduce the resources needed for security, a new scheme with a proxy signature is proposed. If an IoT application adopts the scheme and assigns its signing rights to the underlying nodes, the storage of their public keys is no longer necessary as the application trusts the signing rights. The development of computing technology introduces many risks to cryptographic schemes with a traditional digital signature, and a system with improved security at the quantum-resistant level is required. We propose a scheme with lattice-based cryptography using the bonsai-tree technique for a security system. The new scheme effectively protects the system from a chosen static message attack in the standard model. In the new scheme, an IoT application securely recognizes its honest nodes in the underlying sensor networks at a low storage cost, even in quantum computing.