  Two protocols of quantum direct communication with authentication [Phys. Rev.
A {\bf 73}, 042305 (2006)] are recently proposed by Lee, Lim and Yang. In this
paper we will show that in the two protocols the authenticator Trent should be
prevented from knowing the secret message of communication. The first protocol
can be eavesdropped by Trent using the the intercept-measure-resend attack,
while the second protocol can be eavesdropped by Trent using single-qubit
measurement. To fix these leaks, I revise the original versions of the
protocols by using the Pauli-Z operation $\sigma_z$ instead of the original
bit-flip operation $X$. As a consequence, the protocol securities are improved.
