The identity-based signature (IBS) scheme proposed by Sakai, Ohgishi and Kasahara in 2000, which we refer to as the SOK-IBS scheme, is the first pairing-based IBS scheme. Though most other existing IBS schemes, especially two modified SOK-IBS schemes, have already been proved secure recently, the security of the original SOKIBS scheme is still unclear. In this paper, we prove that the original SOK-IBS scheme is existentially unforgeable under chosen message attacks in the random oracle model. We also show that it is not strongly existentially unforgeable under chosen message attacks, though the two modified versions of it are.