Block lattice polly cracker, theory and practice

We shortly review the Polly Cracker family of cryptosystems. Apparently all the cryptosystems of this family have not resisted even the simplest attacks, through more or less clever linear algebra, as predicted in \cite{barkee}. See \cite{polly} for an update. We review and extend these attacks, and show that the binomial ideals, and especially the lattice ideals, as skecthed in \cite{gb-issac}, might provide secure cryptosystems, also in comparison with other lattice cryptosystems recently proposed