We study block cipher modes that turn a block cipher into a tweakable block cipher, which accepts an auxiliary variable called tweak in addition to the key and message. Liskov et al. first showed such a mode using two keys, where one is the block cipher's key and the other is used for some non-cryptographic function. Later, Rogaway proposed the XEX mode to reduce these two keys to one key. In this paper, we propose a generalization of the Liskov et al.'s scheme with a concrete security proof. Using this, we provide an improved security proof of the XEX and some improvements to the LRW-AES, which is a straightforward AES-based instantiation of Liskov et al.'s scheme proposed by the IEEE Security in Storage Workgroup.
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