We study on the security proof of the improved efficient-Rabin (ERabin) scheme and the F-FDHS scheme. First, we show that the security theorem of the improved ERabin scheme is not correct, and then provide a correct theorem for it. Second, we show that the security theorem of the F-FDHS scheme lacks an assumption. Finally, we present a way to modify the improved ERabin scheme and the F-FDHS scheme.