Cryptographic permutations can be used as a building block in the design of various cryptosystems. They are particularly attractive in bootstrapping a cryptosystem, for example being used in key scheduling and state initialization. Theoretically, permutations can be performed at the bit, byte, or multi-byte integer level. In practice, however, a permutation of two bits has little application value and a permutation of multi-byte integers consumes too much storage, thus only leaving permutations of bytes on the table. In this paper, we analyze and compare three cryptographic permutation schemes, with the first two from the research literature, and the last one introduced in this paper. Different from existing permutation schemes that use slow byte operations for shuffling the permutation table, the newly introduced scheme uses more efficient multi-byte integer operations to achieve the same. Specifically, the new scheme uses byte masks to pseudorandomly extract bytes from a set of integers that are also pseudorandomly selected and then moves extracted bytes from one integer to another using integer operations. To the best of our knowledge, this is the most efficient publicly known cryptographic permutation scheme to date.