Imprimitive Permutation Groups and Trapdoors in Iterated Block Ciphers

An iterated block cipher can be regarded as a means of producing a set of permutations of a message space. Some properties of the group generated by the round functions of such a cipher are known to be of cryptanalytic interest. It is shown here that if this group acts imprimitively on the message space then there is an exploitable weakness in the cipher. It is demonstrated that a weakness of this type can be used to construct a trapdoor that may be difficult to detect. An example of a DES-like cipher, resistant to both linear and differential cryptanalysis that generates an imprimitive group and is easily broken, is given. Some implications for block cipher design are noted.

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