Galindo and Garcia introduced a light-wight identity-based signature (IBS) based on Schnorr signature in AFRICACRYPT 2009. Despite the simple construction and the ef-ficiency of Galindo-Garcia IBS, to the best of our knowledge, their followers do not appear except the IBS scheme by Fukumitsu and Hasegawa in CANDAR'17. Although their IBS achieves the tight security, the efficiency is worse than Galindo-Garcia IBS. Their result suggests that Galindo-Garcia IBS utilizes some special properties of Schnorr signature, and we have to deeply observe the structures of both Galindo-Garcia IBS and Schnorr signature to construct their followers. In this paper, we aim to generalize Galindo-Garcia IBS and to search necessary properties of building cryptographic primitives. As a result, we introduce a generic construction of an IBS from a sigma protocol, and find necessary properties to prove the security of the resulting IBS.