Lattice basis reduction: improved practical algorithms and solving subset sum problems
Mathematical Programming: Series A and B
Hilbert functions and the Buchberger algorithm
Journal of Symbolic Computation
Public-Key Cryptosystems from Lattice Reduction Problems
CRYPTO '97 Proceedings of the 17th Annual International Cryptology Conference on Advances in Cryptology
NTRU: A Ring-Based Public Key Cryptosystem
ANTS-III Proceedings of the Third International Symposium on Algorithmic Number Theory
EUROCRYPT'97 Proceedings of the 16th annual international conference on Theory and application of cryptographic techniques
NTRUSign: digital signatures using the NTRU lattice
CT-RSA'03 Proceedings of the 2003 RSA conference on The cryptographers' track
A digital signature scheme based on CV P∞
PKC'08 Proceedings of the Practice and theory in public key cryptography, 11th international conference on Public key cryptography
Interactions between computer algebra (Gröbner bases) and cryptology
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
Lattice Polly Cracker cryptosystems
Journal of Symbolic Computation
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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.