Extremal Betti Numbers and Applications to Monomial Ideals

In this short note we introduce a notion of extremality for Betti numbers of a minimal free resolution, which can be seen as a refinement of the notion of Mumford-Castelnuovo regularity. We show that extremal Betti numbers of an arbitrary submodule of a free S-module are preserved when taking the generic initial module. We relate extremal multigraded Betti numbers in the minimal resolution of a square free monomial ideal with those of the monomial ideal corresponding to the Alexander dual simplicial complex and generalize theorems of Eagon-Reiner and Terai. As an application we give easy (alternative) proofs of classical criteria due to Hochster, Reisner, and Stanley.Comment: Minor revision. 15 pages, Plain TeX with epsf.tex, 8 PostScript figures, PostScript file available also at http://www.math.columbia.edu/~psorin/eprints/monbetti.p

Graph curves

AbstractWe study a family of stable curves defined combinatorially from a trivalent graph. Most of our results are related to the conjecture of Green which relates the Clifford index of a smooth curve, an important intrinsic invariant measuring the “specialness” of the geometry of the curve, to the “resolution Clifford index,” a projective invariant defined from the canonical embedding. Thus we study the canonical linear series and its powers and identify them in terms of combinatorial data on the graph; we given combinatorial criteria for the canonical series to be base point free or very ample; we prove the analogue of Noether's theorem on the projective normality of smooth canonical curves; we define a combinatorial invariant of a graph which we conjecture to be equal to the resolution Clifford index of the associated graph curve, at least for “most” graphs; and we prove our conjecture for planar graphs and for graphs of Clifford index 0. Along the way we prove a result of some independent interest on the canonical sheaves of (not necessarily arithmetically Cohen-Macaulay) face varieties. The Appendix establishes a formula connecting the combinatorics of a trivalent graph G and the minimal degree of an admissible covering of a curve of arithmetic genus 0 by the corresponding graph curve

Computation of Hilbert functions

AbstractWe present an algorithm along with implementation details and timing data for computing the Hilbert function of a monomial ideal. Our algorithm is often substantially faster in practice than existing algorithms, and executes in linear time when applied to an initial monomial ideal in generic coordinates. The algorithm generalizes to compute multi-graded Hilbert functions

PROPYLA: Privacy Preserving Long-Term Secure Storage

An increasing amount of sensitive information today is stored electronically and a substantial part of this information (e.g., health records, tax data, legal documents) must be retained over long time periods (e.g., several decades or even centuries). When sensitive data is stored, then integrity and confidentiality must be protected to ensure reliability and privacy. Commonly used cryptographic schemes, however, are not designed for protecting data over such long time periods. Recently, the first storage architecture combining long-term integrity with long-term confidentiality protection was proposed (AsiaCCS'17). However, the architecture only deals with a simplified storage scenario where parts of the stored data cannot be accessed and verified individually. If this is allowed, however, not only the data content itself, but also the access pattern to the data (i.e., the information which data items are accessed at which times) may be sensitive information. Here we present the first long-term secure storage architecture that provides long-term access pattern hiding security in addition to long-term integrity and long-term confidentiality protection. To achieve this, we combine information-theoretic secret sharing, renewable timestamps, and renewable commitments with an information-theoretic oblivious random access machine. Our performance analysis of the proposed architecture shows that achieving long-term integrity, confidentiality, and access pattern hiding security is feasible.Comment: Few changes have been made compared to proceedings versio