From the Publisher:
A valuable reference for the novice as well as for the expert who needs a wider scope of coverage within the area of cryptography, this book provides easy and rapid access of information and includes more than 200 algorithms and protocols; more than 200 tables and figures; more than 1,000 numbered definitions, facts, examples, notes, and remarks; and over 1,250 significant references, including brief comments on each paper.

Handbook of Applied Cryptography

Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra. Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis. It can serve as a graduate-level textbook and will be of interest to applied mathematicians, engineers, and computer scientists.

http://www.eeci-institute.eu/GSC2011/Photos-EECI/EECI-GSC-2011-M5/book_AMS.pdf

Optimization Algorithms on Matrix Manifolds

This book is an introduction to the modern approach to the theory of Markov chains. The main goal of this approach is to determine the rate of convergence of a Markov chain to the stationary distribution as a function of the size and geometry of the state space. The authors develop the key tools for estimating convergence times, including coupling, strong stationary times, and spectral methods. Whenever possible, probabilistic methods are emphasized. The book includes many examples and provides brief introductions to some central models of statistical mechanics. Also provided are accounts of random walks on networks, including hitting and cover times, and analyses of several methods of shuffling cards. As a prerequisite, the authors assume a modest understanding of probability theory and linear algebra at an undergraduate level. ""Markov Chains and Mixing Times"" is meant to bring the excitement of this active area of research to a wide audience.

Markov Chains and Mixing Times

Monomial Ideals.- Squarefree monomial ideals.- Borel-fixed monomial ideals.- Three-dimensional staircases.- Cellular resolutions.- Alexander duality.- Generic monomial ideals.- Toric Algebra.- Semigroup rings.- Multigraded polynomial rings.- Syzygies of lattice ideals.- Toric varieties.- Irreducible and injective resolutions.- Ehrhart polynomials.- Local cohomology.- Determinants.- Plucker coordinates.- Matrix Schubert varieties.- Antidiagonal initial ideals.- Minors in matrix products.- Hilbert schemes of points.

https://www.springer.com/productFlyer_978-0-387-22356-8.pdf?SGWID=0-0-1297-34952457-0

Combinatorial Commutative Algebra

Preface. Part 1: Basic Theory - The Simplex Method and Duality. 1. Introduction. 2. The Simplex Method. 3. Degeneracy. 4. Efficiency of the Simplex Method. 5. Duality Theory. 6. The Simplex Method in Matrix Notation. 7. Sensitivity and Parametric Analyses. 8. Implementation Issues. 9. Problems in General Form. 10. Convex Analysis. 11. Game Theory. 12. Regression. Part 2: Network-Type Problems. 13. Network Flow Problems. 14. Applications. 15. Structural Optimization. Part 3: Interior-Point Methods. 16. The Central Path. 17. A Path-Following Method. 18. The KKT System. 19. Implementation Issues. 20. The Affine-Scaling Method. 21. The Homogeneous Self-Dual Method. Part 4: Extensions. 22. Integer Programming. 23. Quadratic Programming. 24. Convex Programming. Appendix A: Source Listings. Answers to Selected Exercises. Bibliography. Index.

/pdf/linear-programming-foundations-and-extensions-5kn95li7g9.pdf

Linear Programming: Foundations and Extensions

We analyze the most commonly used method for shuffling cards. The main result is a simple expression for the chance of any arrangement after any number of shuffles. This is used to give sharp bounds on the approach to randomness: $\frac{3}{2} \log_2 n + \theta$ shuffles are necessary and sufficient to mix up $n$ cards. Key ingredients are the analysis of a card trick and the determination of the idempotents of a natural commutative subalgebra in the symmetric group algebra.

/pdf/trailing-the-dovetail-shuffle-to-its-lair-54vfoxd1jm.pdf

Trailing the Dovetail Shuffle to its Lair

To establish that a document was created after a given moment in time, it is necessary to report events that could not have been predicted before they happened. To establish that a document was created before a given moment in time, it is necessary to cause an event based on the document, which can be observed by others. Cryptographic hash functions can be used both to report events succinctly, and to cause events based on documents without revealing their contents. Haber and Stornetta have proposed two schemes for digital time-stamping which rely on these principles [HaSt 91].

Improving the Efficiency and Reliability of Digital Time-Stamping

This paper evolved from a long series of discussions between the two authors, going backto around 1980, on the problems of mak ing effective computations in algebraic geometry, and it tookmore definite shape in a survey talkgiven by the second author at a conference on Computer Algebra in 1984. The goal at that time was to bring together the perspectives of theoretical computer scientists and of working algebraic geometers, while laying out what we considered to be the main computational problems and bounds on their complexity. Only part of the talkwas written down and since that time there has been a good deal of progress. However, the material that was written up may still serve as a useful introduction to some of the ideas and estimates used in this field (at least the editors of this volume thinkso), even though most of the results included here are either published elsewhere, or exist as “folk-theorems” by now. The article has four sections. The first two parts are concerned with

/pdf/what-can-be-computed-in-algebraic-geometry-28xv6tw8je.pdf

What can be computed in algebraic geometry

We construct a canonical free resolution for arbitrary monomial modules and lattice ideals. This includes monomial ideals and defining ideals of toric varieties, and it generalizes our joint results with Irena Peeva for generic ideals [BPS],[PS]. Introduction Given a field k, we consider the Laurent polynomial ring T = k[x±1 1 , . . . , x ±1 n ] as a module over the polynomial ring S = k[x1, . . . , xn]. The module structure comes from the natural inclusion of semigroup algebras S = k[N] ⊂ k[Z] = T . A monomial module is an S-submodule of T which is generated by monomials x = x1 1 · · ·xan n , a ∈ Z. Of special interest are the two cases when M has a minimal monomial generating set which is either finite or forms a group under multiplication. In the first case M is isomorphic to a monomial ideal in S. In the second case M coincides with the lattice module ML := S {x | a ∈ L} = k {x | b ∈ N + L} ⊂ T. for some sublattice L ⊂ Z whose intersection with N is the origin 0 = (0, . . . , 0). We shall derive free resolutions of M from regular cell complexes whose vertices are the generators of M and whose faces are labeled by the least common multiples of their vertices. The basic theory of such cellular resolutions is developed in Section 1. Our main result is the construction of the hull resolution in Section 2. We rescale the exponents of the monomials in M , so that their convex hull in R is a polyhedron Pt whose bounded faces support a free resolution of M . This resolution is new and interesting even for monomial ideals. It need not be minimal, but, unlike minimal resolutions, it respects symmetry and is free from arbitrary choices. In Section 3 we relate the lattice module ML to the Z/L-graded lattice ideal IL = 〈 x − x | a− b ∈ L 〉 ⊂ S. This class of ideals includes ideals defining toric varieties. We express the cyclic Smodule S/IL as the quotient of the infinitely generated S-module ML by the action of L. In fact, we like to think of ML as the “universal cover” of IL. Many questions about IL can thus be reduced to questions about ML. In particular, we obtain the hull resolution of a lattice ideal IL by taking the hull resolution of ML modulo L. This paper is inspired by the work of Barany, Howe and Scarf [BHS] who introduced the polyhedron Pt in the context of integer programming. The hull resolution generalizes results in [BPS] for generic monomial ideals and in [PS] for generic lattice ideals. In these generic cases the hull resolution is minimal.

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Cellular Resolutions of Monomial Modules

Call a monomial ideal M "generic" if no variable appears with the same nonzero exponent in two distinct monomial generators Using a convex polytope first studied by Scarf, we obtain a minimal free resolution of M Any monomial ideal M can be made generic by deformation of its generating exponents Thus, the above construction yields a (usually nonminimal) resolution of M for arbitrary monomial ideals, bounding the Betti numbers of M in terms of the Upper Bound Theorem for Convex Polytopes We show that our resolutions are DG-algebras, and consider realizability questions and irreducible decompositions

Dave Bayer

Papers

Trailing the Dovetail Shuffle to its Lair

Improving the Efficiency and Reliability of Digital Time-Stamping

What can be computed in algebraic geometry

Cellular Resolutions of Monomial Modules

Monomial Resolutions