In the model, a sender S wants to send a message to a receiver R secretly and reliably in r-round. They do not share any information like keys, but there are it independent communication channels between S and R, and an adversary A can observe and/or substitute the data which goes through some channels (but not all). In this paper, we propose almost secure (1-round, 3t + 1-channel) MTSs which have following two properties where t is the number of channels A can observe and/or forge. (1) The running time of message decryption algorithm is polynomial in it. (2) Communication cost is smaller than the previous MTSs, if the message is large to some degree.