A smart meter, a component of smart grid systems, has low computational capability. This is in contrast to data collection units (DCUs) and meter data management servers (MDMSs). In this study, we propose a lightweight signature scheme and an authentication and report protocol that can reduce computational overhead and facilitate efficient operation of these three components in smart grid systems. The proposed signature scheme, called the linear map digital signature scheme (LMDSS), uses properties of linear maps and matrix operations; hence, it has low computational requirements. The proposed protocol is a delegation-based authentication protocol that uses an intermediate node, DCU. Using the proposed protocol, authentication of a DCU can be completed by a hash function. To evaluate the performance of the scheme and protocol, we implemented signature schemes, ours and others, and simulated the proposed protocol to obtain analytical results. We also proved that the scheme is secure by using a random oracle and analyzing the security of the protocol based on possible attack scenarios.