  In this paper we consider cryptographic applications of the arithmetic on the
hyperoctahedral group. On an appropriate subgroup of the latter, we
particularly propose to construct public key cryptosystems based on the
discrete logarithm. The fact that the group of signed permutations has rich
properties provides fast and easy implementation and makes these systems
resistant to attacks like the Pohlig-Hellman algorithm. The only negative point
is that storing and transmitting permutations need large memory. Using together
the hyperoctahedral enumeration system and what is called subexceedant
functions, we define a one-to-one correspondance between natural numbers and
signed permutations with which we label the message units.
