Private Simultaneous Messages Based on Quadratic Residues

Private Simultaneous Messages (PSM) model is a minimal model for secure multiparty computation. Feige, Kilian, and Naor (STOC 1994) and Ishai (Cryptology and Information Security Series 2013) constructed PSM protocols based on quadratic residues. In this paper, we define QR-PSM protocols as a generalization of these protocols. A QR-PSM protocol is a PSM protocol whose decoding function outputs the quadratic residuosity of what is computed from messages. We design a QR-PSM protocol for any symmetric function $f: \{0,1\}^n \rightarrow \{0,1\}$ of communication complexity $O(n^2)$. As far as we know, it is the most efficient PSM protocol since the previously known best PSM protocol was of $O(n^2\log n)$ (Beimel et al., CRYPTO 2014). We also study the sizes of the underlying finite fields $\mathbb{F}_p$ in the protocols since the communication complexity of a QR-PSM protocol is proportional to the bit length of the prime $p$. In particular, we show that the $N$-th Peralta prime $P_N$, which is used for general QR-PSM protocols, can be taken as at most $(1+o(1))N^2 2^{2N-2}$, which improves the Peralta's known result (Mathematics of Computation 1992) by a constant factor $(1+\sqrt{2})^2$

Ample simplicial complexes

Motivated by potential applications in network theory, engineering and computer science, we study $r$-ample simplicial complexes. These complexes can be viewed as finite approximations to the Rado complex which has a remarkable property of {\it indestructibility,} in the sense that removing any finite number of its simplexes leaves a complex isomorphic to itself. We prove that an $r$-ample simplicial complex is simply connected and $2$-connected for $r$ large. The number $n$ of vertexes of an $r$-ample simplicial complex satisfies $\exp(\Omega(\frac{2^r}{\sqrt{r}}))$. We use the probabilistic method to establish the existence of $r$-ample simplicial complexes with $n$ vertexes for any $n>r 2^r 2^{2^r}$. Finally, we introduce the iterated Paley simplicial complexes, which are explicitly constructed $r$-ample simplicial complexes with nearly optimal number of vertexes

Graph Theory

Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

Combinatorics

This is the report on the Oberwolfach workshop on Combinatorics, held 1–7 January 2006. Combinatorics is a branch of mathematics studying families of mainly, but not exclusively, finite or countable structures – discrete objects. The discrete objects considered in the workshop were graphs, set systems, discrete geometries, and matrices. The programme consisted of 15 invited lectures, 18 contributed talks, and a problem session focusing on recent developments in graph theory, coding theory, discrete geometry, extremal combinatorics, Ramsey theory, theoretical computer science, and probabilistic combinatorics

Homomorphisms of (j,k)-mixed graphs

A mixed graph is a simple graph in which a subset of the edges have been assigned directions to form arcs. For non-negative integers j and k, a (j,k)−mixed graph is a mixed graph with j types of arcs and k types of edges. The collection of (j,k)−mixed graphs contains simple graphs ((0,1)−mixed graphs), oriented graphs ((1,0)−mixed graphs) and k−edge- coloured graphs ((0,k)−mixed graphs).A homomorphism is a vertex mapping from one (j,k)−mixed graph to another in which edge type is preserved, and arc type and direction are preserved. The (j,k)−chromatic number of a (j,k)−mixed graph is the least m such that an m−colouring exists. When (j,k)=(0,1), we see that these definitions are consistent with the usual definitions of graph homomorphism and graph colouring.In this thesis we study the (j,k)−chromatic number and related parameters for different families of graphs, focussing particularly on the (1,0)−chromatic number, more commonly called the oriented chromatic number, and the (0,k)−chromatic number.In addition to considering vertex colourings, we also consider incidence colourings of both graphs and digraphs. Using systems of distinct representatives, we provide a new characterisation of the incidence chromatic number. We define the oriented incidence chromatic number and find, by way of digraph homomorphism, a connection between the oriented incidence chromatic number and the chromatic number of the underlying graph. This connection motivates our study of the oriented incidence chromatic number of symmetric complete digraphs.Un graphe mixte est un graphe simple tel que un sous-ensemble des arêtes a une orientation. Pour entiers non négatifs j et k, un graphe mixte-(j,k) est un graphe mixte avec j types des arcs and k types des arêtes. La famille de graphes mixte-(j,k) contient graphes simple, (graphes mixte−(0,1)), graphes orienté (graphes mixte−(1,0)) and graphe coloré arête −k (graphes mixte−(0,k)).Un homomorphisme est un application sommet entre graphes mixte−(j,k) que tel les types des arêtes sont conservés et les types des arcs et leurs directions sont conservés. Le nombre chromatique−(j,k) d’un graphe mixte−(j,k) est le moins entier m tel qu’il existe un homomorphisme à une cible avec m sommets. Quand on observe le cas de (j,k) = (0,1), on peut déterminer ces définitions correspondent à les définitions usuel pour les graphes.Dans ce mémoire on etude le nombre chromatique−(j,k) et des paramètres similaires pour diverses familles des graphes. Aussi on etude les coloration incidence pour graphes and digraphs. On utilise systèmes de représentants distincts et donne une nouvelle caractérisation du nombre chromatique incidence. On define le nombre chromatique incidence orienté et trouves un connexion entre le nombre chromatique incidence orienté et le nombre chromatic du graphe sous-jacent

The Register, 1970-03-06

https://digital.library.ncat.edu/atregister/1380/thumbnail.jp

The Register, 1970-03-06

https://digital.library.ncat.edu/atregister/1380/thumbnail.jp

From Large to In nite Random Simplicial Complexes.

PhD ThesesRandom simplicial complexes are a natural higher dimensional generalisation to the models of random graphs from Erd}os and R enyi of the early 60s. Now any topological question one may like to ask raises a question in probability - i.e. what is the chance this topological property occurs? Several models of random simplicial complexes have been intensely studied since the early 00s. This thesis introduces and studies two general models of random simplicial complexes that includes many well-studied models as a special case. We study their connectivity and Betti numbers, prove a satisfying duality relation between the two models, and use this to get a range of results for free in the case where all probability parameters involved are uniformly bounded. We also investigate what happens when we move to in nite dimensional random complexes and obtain a simplicial generalisation of the Rado graph, that is we show the surprising result that (under a large range of parameters) every in nite random simplicial complexes is isomorphic to a given countable complex X with probability one. We show that this X is in fact homeomorphic to the countably in nite ball. Finally, we look at and construct nite approximations to this complex X, and study their topological properties