8.8 Cryptosystems

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.