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A general result is proved for constructions which use a pseudo-random function (PRF) with a “small” domain to build a PRF with a “large” domain. This result is used to analyse a new block-cipher based parallelizable PRF, called iPMAC which improves upon the well-known PMAC algorithm. New authenticated encryption schemes are described and then combined with iPMAC to obtain new schemes for authenticated encryption with associated data. Improvements over well known schemes such as the offset codebook (OCB) mode include avoiding a design-stage discrete logarithm computation, a small speed-up and a smaller size decryption algorithm.