General error locator polynomials were introduced in 2005 as an alternative decoding for cyclic codes. We now present a conjecture on their sparsity, which would imply polynomial-time decoding for all cyclic codes. A general result on the explicit form of the general error locator polynomial for all cyclic codes is given, along with several results for specific code families, providing evidence to our conjecture. From these, a theoretical justification of the sparsity of general error locator polynomials is obtained for all binary cyclic codes with t ≤ 2 and n <; 105, as well as for t = 3 and n <; 63, except for some cases where the conjectured sparsity is proved by a computer check. Moreover, we summarize all related results, previously published, and we show how they provide further evidence to our conjecture. Finally, we discuss the link between our conjecture and the complexity of bounded-distance decoding of the cyclic codes.

Up to now, any attempt to use Gröbner bases in the design of public key cryptosystems has failed, as anticipated by a classical paper of B. Barkee et al.; we show why, and show that the only residual hope is to use binomial ideals, i.e. lattices. We propose two lattice-based cryptosystems that will show the usefulness of multivariate polynomial algebra and Grobner bases in the construction of public key cryptosystems. The first one tries to revive two cryptosystems Polly Cracker and GGH, that have been considered broken, through a hybrid; the second one improves a cryptosystem (NTRU) that only has heuristic and challenged evidence of security, providing evidence that the extension cannot be broken with some of the standard lattice tools that can be used to break some reduced form of NTRU. Because of the bounds on length, we only sketch the construction of these two cryptosystems, and leave many details of the construction of private and public keys, of the proofs and of the security considerations to forthcoming technical papers.