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A binary relation on a finite set is called a Hall relation if it contains a permutation of the set. Under the usual relational product, Hall relations form a semigroup which is known to be a block-group, that is, a semigroup with at most one idempotent in each $mathrsfs{R}$-class and each $mathrsfs{L}$-class. Here we show that in a certain sense, the converse is true: every block-group divides a semigroup of Hall relations on a finite set.
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The group generated by the round functions of a block ciphers is a widely investigated problem. We identify a large class of block ciphers for which such group is easily guaranteed to be primitive. Our class includes the AES and the SERPENT.
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