This is a study on a certain group theoretic property of the set of encrypt
ion functions of a block cipher. We have shown how to construct a subset wh
ich has this property in a given symmetric group by a computer algebra soft
ware GAP4.2 (Groups, Algorithms, and Programming, Version 4.2). These obser
vations on group structures of block ciphers suggest us that we may be able
to set a trapdoor based on meet-in-the-middle attack on block ciphers.