<p>A linearly homomorphic signature (LHS) enables linear computation on signed data and has been investigated in many contexts, such as network coding to resist pollution attacks and computation on outsourced data. Traditional LHS security relies entirely on the secrecy of the signing keys. The exposure of signing keys necessitates updating all generated signatures. However, with the increasing use of relatively insecure mobile devices in network coding and data outsourcing systems, key exposure has become more prevalent. To mitigate the hazard of key exposure in the LHS setting, we integrate key update mechanisms into the LHS by presenting a forward-secure linearly homomorphic signature (FSLHS). Specifically, we formalize the definition and security notions for the FSLHS scheme and present a concrete implementation. We prove our scheme to be forward secure against adaptively chosen message attacks, assuming the hardness of the CDH problem. Compared to previous works, our performance analysis shows that all parameters of our scheme exhibit logarithmic complexity with respect to the total number of time periods.</p>