In cryptography, secure channels enable the confidential and authenticated message exchange between authorized users. A generic approach of constructing such channels is by combining an encryption primitive with an authentication primitive (MAC). In this work, we introduce the design of a new cryptographic primitive to be used in the construction of secure channels. Instead of using general purpose MACs, we propose the employment of special purpose MACs, named "epsilon-MACs". The main motive behind this work is the observation that, since the message must be both encrypted and authenticated, there can be a redundancy in the computations performed by the two primitives. If this turned out to be the case, removing such redundancy will improve the efficiency of the overall construction. In addition, computations performed by the encryption algorithm can be further utilized to improve the security of the authentication algorithm. In this work, we show how epsilon-MACs can be designed to reduce the amount of computations required by standard MACs based on universal hash functions, and show how epsilon-MACs can be secured against key-recovery attacks.