Results 1  10
of
52
Compact Proofs of Retrievability
, 2008
"... In a proofofretrievability system, a data storage center must prove to a verifier that he is actually storing all of a client’s data. The central challenge is to build systems that are both efficient and provably secure — that is, it should be possible to extract the client’s data from any prover ..."
Abstract

Cited by 72 (0 self)
 Add to MetaCart
In a proofofretrievability system, a data storage center must prove to a verifier that he is actually storing all of a client’s data. The central challenge is to build systems that are both efficient and provably secure — that is, it should be possible to extract the client’s data from any prover that passes a verification check. All previous provably secure solutions require that a prover send O(l) authenticator values (i.e., MACs or signatures) to verify a file, for a total of O(l 2) bits of communication, where l is the security parameter. The extra cost over the ideal O(l) communication can be prohibitive in systems where a verifier needs to check many files. We create the first compact and provably secure proof of retrievability systems. Our solutions allow for compact proofs with just one authenticator value — in practice this can lead to proofs with as little as 40 bytes of communication. We present two solutions with similar structure. The first one is privately verifiable and builds elegantly on pseudorandom functions (PRFs); the second allows for publicly verifiable proofs and is built from the signature scheme of Boneh, Lynn, and Shacham in bilinear groups. Both solutions rely on homomorphic properties to aggregate a proof into one small authenticator value. 1
Datatypegeneric programming
 Spring School on DatatypeGeneric Programming, volume 4719 of Lecture Notes in Computer Science
"... Abstract. Generic programming aims to increase the flexibility of programming languages, by expanding the possibilities for parametrization — ideally, without also expanding the possibilities for uncaught errors. The term means different things to different people: parametric polymorphism, data abst ..."
Abstract

Cited by 48 (12 self)
 Add to MetaCart
Abstract. Generic programming aims to increase the flexibility of programming languages, by expanding the possibilities for parametrization — ideally, without also expanding the possibilities for uncaught errors. The term means different things to different people: parametric polymorphism, data abstraction, metaprogramming, and so on. We use it to mean polytypism, that is, parametrization by the shape of data structures rather than their contents. To avoid confusion with other uses, we have coined the qualified term datatypegeneric programming for this purpose. In these lecture notes, we expand on the definition of datatypegeneric programming, and present some examples of datatypegeneric programs. We also explore the connection with design patterns in objectoriented programming; in particular, we argue that certain design patterns are just higherorder datatypegeneric programs. 1
Partition bijections, a survey
 Ramanujan J
"... Abstract. We present an extensive survey of bijective proofs of classical partitions identities. While most bijections are known, they are often presented in a different, sometimes unrecognizable way. Various extensions and generalizations are added in the form of exercises. ..."
Abstract

Cited by 37 (12 self)
 Add to MetaCart
Abstract. We present an extensive survey of bijective proofs of classical partitions identities. While most bijections are known, they are often presented in a different, sometimes unrecognizable way. Various extensions and generalizations are added in the form of exercises.
On the rank of a tropical matrix
"... Abstract. This is a foundational paper in tropical linear algebra, which is linear algebra over the minplus semiring. We introduce and compare three natural definitions of the rank of a matrix, called the Barvinok rank, the Kapranov rank and the tropical rank. We demonstrate how these notions arise ..."
Abstract

Cited by 24 (5 self)
 Add to MetaCart
Abstract. This is a foundational paper in tropical linear algebra, which is linear algebra over the minplus semiring. We introduce and compare three natural definitions of the rank of a matrix, called the Barvinok rank, the Kapranov rank and the tropical rank. We demonstrate how these notions arise naturally in polyhedral and algebraic geometry, and we show that they differ in general. Realizability of matroids plays a crucial role here. Connections to optimization are also discussed. 1.
Design Patterns as HigherOrder DatatypeGeneric Programs
, 2006
"... Design patterns are reusable abstractions in objectoriented software. However, using current mainstream programming languages, these elements can only be expressed extralinguistically: as prose, pictures, and prototypes. We believe that this is not inherent in the patterns themselves, but evidence ..."
Abstract

Cited by 14 (6 self)
 Add to MetaCart
Design patterns are reusable abstractions in objectoriented software. However, using current mainstream programming languages, these elements can only be expressed extralinguistically: as prose, pictures, and prototypes. We believe that this is not inherent in the patterns themselves, but evidence of a lack of expressivity in the languages of today. We expect that, in the languages of the future, the code parts of design patterns will be expressible as reusable library components. Indeed, we claim that the languages of tomorrow will suffice; the future is not far away. All that is needed, in addition to commonlyavailable features, are higherorder and datatypegeneric constructs; these features are already or nearly available now. We argue the case by presenting higherorder datatypegeneric programs capturing ORIGAMI, a small suite of patterns for recursive data structures.
Planar decompositions and the crossing number of graphs with an excluded minor
 IN GRAPH DRAWING 2006; LECTURE NOTES IN COMPUTER SCIENCE 4372
, 2007
"... Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar decomposition with bounded width and a linear number of bags. The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. We prove that planar decompositions are intimately related to the crossing number. In particular, a graph with bounded degree has linear crossing number if and only if it has a planar decomposition with bounded width and linear order. It follows from the above result about planar decompositions that every graph with bounded degree and an excluded minor has linear crossing number. Analogous results are proved for the convex and rectilinear crossing numbers. In particular, every graph with bounded degree and bounded treewidth has linear convex crossing number, and every K3,3minorfree graph with bounded degree has linear rectilinear crossing number.
Divisible designs, Laguerre geometry, and beyond. Quaderni Sem
"... 5.3 Divisible designs vs. finite chain geometries........... 64 ..."
Abstract

Cited by 12 (5 self)
 Add to MetaCart
5.3 Divisible designs vs. finite chain geometries........... 64
Solving random satisfiable 3CNF formulas in expected polynomial time
 In Proc. 17th ACMSIAM Symp. on Discrete Algorithms
, 2006
"... We present an algorithm for solving 3SAT instances. Several algorithms have been proved to work whp (with high probability) for various SAT distributions. However, an algorithm that works whp has a drawback. Indeed for typical instances it works well, however for some rare inputs it does not provide ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
We present an algorithm for solving 3SAT instances. Several algorithms have been proved to work whp (with high probability) for various SAT distributions. However, an algorithm that works whp has a drawback. Indeed for typical instances it works well, however for some rare inputs it does not provide a solution at all. Alternatively, one could require that the algorithm always produce a correct answer but perform well on average. Expected polynomial time formalizes this notion. We prove that for some natural distribution on 3CNF formulas, called planted 3SAT, our algorithm has expected polynomial (in fact, almost linear) running time. The planted 3SAT distribution is the set of satisfiable 3CNF formulas generated in the following manner. First, a truth assignment is picked uniformly at random. Then, each clause satisfied by it is included in the formula with probability p. Extending previous work for the planted 3SAT distribution, we present, for the first time for a satisfiable SAT distribution, an expected polynomial time algorithm. Namely, it solves all 3SAT instances, and over the planted distribution (with p = d/n 2, d> 0 a sufficiently large constant) it runs in expected polynomial time. Our results extend to kSAT for any constant k.
Andreev’s theorem on hyperbolic polyhedra
, 2006
"... In 1970, E. M. Andreev published a classification of all threedimensional compact hyperbolic polyhedra having nonobtuse dihedral angles [3, 4]. Given a combinatorial description of a polyhedron, C, Andreev’s Theorem provides five classes of linear inequalities, depending on C, for the dihedral ang ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
In 1970, E. M. Andreev published a classification of all threedimensional compact hyperbolic polyhedra having nonobtuse dihedral angles [3, 4]. Given a combinatorial description of a polyhedron, C, Andreev’s Theorem provides five classes of linear inequalities, depending on C, for the dihedral angles, which are necessary and sufficient conditions for the existence of a hyperbolic polyhedron realizing C with the assigned dihedral angles. Andreev’s Theorem also shows that the resulting polyhedron is unique, up to hyperbolic isometry. Andreev’s Theorem is both an interesting statement about the geometry of hyperbolic 3dimensional space, as well as a fundamental tool used in the proof for Thurston’s Hyperbolization Theorem for 3dimensional Haken manifolds. It is also remarkable to what level the proof of Andreev’s Theorem resembles (in a simpler way) the proof of Thurston. We correct a fundamental error in Andreev’s proof of existence and also provide a readable new proof of the other parts of the proof of Andreev’s Theorem, because
The journey of the four colour theorem through time
 The NZ Math. Magazine
"... This is a historical survey of the Four Colour Theorem and a discussion of the philosophical implications of its proof. The problem, first stated as far back as 1850s, still causes controversy today. Its computeraided proof has forced mathematicians to question the notions of proofs and mathematical ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
This is a historical survey of the Four Colour Theorem and a discussion of the philosophical implications of its proof. The problem, first stated as far back as 1850s, still causes controversy today. Its computeraided proof has forced mathematicians to question the notions of proofs and mathematical truth.