In CRYPTO 1995, Bellare, $Gu'{e}rin$, and Rogaway proposed a very efficient message authentication scheme. This scheme is secure against adaptive chosen message attacks, under the assumption that its underlying primitive is a pseudorandom function. This article studies how to weaken that assumption. For an adaptive chosen message attack, we take into account two scenarios. On the one hand, the adversary intercepts the authenticated messages corresponding to messages chosen adaptively by herself, so the verifier does not receive them. On the other hand, the adversary can only eavesdrop the authenticated messages corresponding to messages chosen adaptively by herself, so the verifier receives them. We modify the original scheme. In the first scenario, our scheme is secure if the underlying primitive is a pseudorandom function. In the second scenario, our scheme is still secure under a weaker assumption that the underlying primitive is an indistinguishable-uniform function.