A study of the high-pressure elastic properties of ideal stoichiometric platinum carbide (PtC) in the rock-salt (RS) and zinc-blende (ZB) structures was conducted using first-principles calculations based on density functional theory, in which we employ the generalized gradient approximation of the Perdew-Burke-Eruzerhof form together with plane-wave basis sets for expanding the crystal orbitals and periodic electron density. Our calculation shows that the recently synthesized compound PtC possess a high-bulk modulus value in the RS phase and the ZB phase is more stable. The investigation of the elastic stability under pressure indicated that the transition pressure from ZB to RS structure of PtC is about 30 GPa and the high-pressure RS phase is stable up to 100 GPa. Our conclusions are consistent with the other theoretical predictions but are reversed with the diamond anvil cell experimental results. Therefore, the experimental observation of the RS structure in PtC remains a puzzle and our study indicates that more experimental and theoretical works need to be performed to ascertain the true nature of the newly discovered PtC material. In addition, the pressure dependence of the bulk modulus K, the shear modulus G, the Young's modulus E, the Poisson's ratio nu, the Debye temperature Theta(D), the compressional wave velocity V-p, the shear wave velocity V-s, and the elastic anisotropy factor A for the ZB and RS structures of PtC are all successfully obtained. Moreover, the pressure dependence of the longitudinal and the shear wave velocities in three directions [100], [110], and [111] for cubic PtC are also predicted for the first time.