This work builds on earlier work by Rogaway at Asiacrypt 2004 on tweakable block cipher (TBC) and modes of operations. Our first contribution is to generalize Rogaway's TBC construction by working over a ring and by the use of a masking sequence of functions. The ring can be instantiated as either GF or as . Further, over GF, efficient instantiations of the masking sequence of functions can be done using either a binary linear feedback shift register (LFSR); a powering construction; a cellular automata map; or by using a word-oriented LFSR. Rogaway's TBC construction was built from the powering construction over GF. Our second contribution is to use the general TBC construction to instantiate constructions of various modes of operations including authenticated encryption (AE) and message authentication code (MAC). In particular, this gives rise to a family of efficient one-pass AE modes of operation. Out of these, the mode of operation obtained by the use of word-oriented LFSR promises to provide a masking method which is more efficient than the one used in the well known AE protocol called OCB1.