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As seen earlier algebraic cryptosystems rely on explicit manipulations
with finite abelian groups. All the finite abelian groups that have
been used as cryptosystems so far are specific *K*-groups of schemes
with minor modifications. Thus it would seem that a useful way of
diversifying the collection of groups available for cryptosystems
would be to study all *K*-groups of schemes. This is difficult because
there is (so far) no way to explicitly bound the generators of such
groups--indeed the fact that these groups are finitely generated is
no yet proved! In computational applications we would also need
explicit ways of representing elements and reducing sums of such
elements to the representative ones. While the description of every
element in terms of matrices using the ``syzygy'' approach described
above is possible much more work needs to be done to make *K*-groups
of all schemes computationally approachable. However, in the case of
some specific schemes this can be done. This is what we explore in the
next section.

Kapil Hari Paranjape
2002-10-20