Currently, the majority of lattice-base identity-based signature (LB-IBS) schemes rely on lattice trapdoor technique introduced by Gentry, Peikert, and Vaikuntanathan. However, this approach significantly impacts computational efficiency. In light of this, we present an alternative method for constructing LB-IBS that does not rely on lattice trapdoors and preimage sampling function. We construct an IBS scheme by employing Lyubashevsky’s signature twice, which is essentially a public key delegation from concatenated Schnorr signatures. In order to optimize the signature size, we describe an adaptation of our scheme in module lattices setting and compression technique. Moreover, our new LB-IBS can combine with Dilithium naturally so that it could carry the advantages of the latter such as side-channel attacks protection. We demonstrate that our construction is provably secure against existential forgery on adaptively chosen message and identity attacks in random oracle model. In comparison with the most efficient LB-IBS scheme (to the best of our knowledge), the signature size of our scheme is 52% smaller and runtime of our scheme is up to 32% faster.