Fabrizio Caruso

Certificates of Positivity in the Bernstein Basis

Abstract Let

Gröbner bases for public key cryptography

P\in\mathbb{Z[X]}

The SARAG library: some algorithms in real algebraic geometry

be a polynomial of degree p with coefficients in the monomial basis of bit-size bounded by τ. If P is positive on [−1,1], we obtain a certificate of positivity (i.e., a description of P making obvious that it is positive) of bit-size O(p4(τ+log 2p)). Previous comparable results had a bit-size complexity exponential in p and τ (Powers and Reznick in Trans. Am. Math. Soc. 352(10):4677–4692, 2000; Powers and Reznick in J. Pure Appl. Algebra 164:221–229, 2001).

Lattice Polly Cracker cryptosystems

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.

Gröbner Bases in Public Key Cryptography

In this paper we present SARAG, which is a software library for real algebraic geometry written in the free computer algebra system Maxima. SARAG stands for “Some Algorithms in Real Algebraic Geometry” and has two main applications: extending the capabilities of Maxima in the field of real algebraic geometry and being part of the interactive version of the book “Algorithms in Real Algebraic Geometry” by S. Basu, R. Pollack, M.-F. Roy, which can be now freely downloaded. The routines of the library deal with: theory of signed sub-resultants, linear algebra, gcd computation, real roots counting, real roots isolation, sign determination, Thom encodings, study of the topology of curves. At the moment SARAG is being used as a tool to develop, implement and tune algorithms coming from new research results, e.g. an algorithm for faster gcd computation, an algorithm for the study of the topology of curves over non-Archimedian real closed fields.