Group signatures are cryptographic primitives where users can anonymously sign messages in the name of a population they belong to. Gordon {\em et al.} (Asiacrypt 2010) suggested the first realization of group signatures based on lattice assumptions in the random oracle model. A significant drawback of their scheme is its linear signature size in the cardinality $N$ of the group. A recent extension proposed by Camenisch {\em et al.} (SCN 2012) suffers from the same overhead. In this paper, we describe the first lattice-based group signature schemes where the signature and public key sizes are essentially logarithmic in~$N$ (for any fixed security level). Our basic construction only satisfies a relaxed definition of anonymity (just like the Gordon {\it et al.} system) but readily extends into a fully anonymous group signature ({\it i.e.}, that resists adversaries equipped with a signature opening oracle). We prove the security of our schemes in the random oracle model under the SIS and LWE assumptions.