Properties Of Catalan Numbers Research Articles

ENCRYPTION OF 3D PLANE IN GIS USING VORONOI-DELAUNAY TRIANGULATIONS AND CATALAN NUMBERS

A method of encryption of the 3D plane in Geographic Information Systems (GIS) is presented. The method is derived using Voronoi-Delaunay triangulation and properties of Catalan numbers. The Voronoi-Delaunay incremental algorithm is presented as one of the most commonly used triangulation techniques for random point selection. In accordance with the multiple applications of Catalan numbers in solving combinatorial problems and their "bit-balanced" characteristic, the process of encrypting and decrypting the coordinates of points using the Lattice Path method (walk on the integer lattice) or LIFO model is given. The triangulation of the plane started using decimal coordinates of a set of given planar points. Afterward, the resulting decimal values of the coordinates are converted to corresponding binary records and the encryption process starts by a random selection of the Catalan key according to the LIFO model. These binary coordinates are again converted into their original decimal values, which enables the process of encrypted triangulation. The original triangulation of the plane can be generated by restarting the triangulation algorithm. Due to its exceptional efficiency in terms of launching programs on various computer architectures and operating systems, Java programming language enables an efficient implementation of our method.

A novel approach to steganography based on the properties of Catalan numbers and Dyck words

Abstract This paper presents a new method of data hiding using Catalan numbers and Dyck words. The proposed steganographic solution belongs to the category of techniques based on the key generating process, rather than steganographic techniques such as injecting or substituting bits. The complex stego key consists of three sets of values that provide the technique of a hidden message generation completely. Hidden message is generated based on the data carrier and an adequate complex stego key. An important characteristic of the proposed method is that the data carrier retains its original shape, without supplements or modifications. Proposed method is explained in detail through the general scenario and through concrete examples. State of the art steganographic analysis of the proposed solution is presented in this paper, as well as possible suggestions for application in business information systems, authentication and distribution of secret cryptographic keys.

Some more properties of Catalan numbers

We want to illustrate some correspondences between Catalan numbers and combinatoric objects, such as plane walks, binary trees and some particular words. By means of under-diagonal walks, we give a combinatorial interpretation of the formula Cn = 1n+12nn defining Catalan numbers. These numbers also enumerate both words in a particular language defined on a four character alphabet and the corresponding walks made up of four different types of steps. We illustrate a bijection between n-long words in this language and binary trees having n + 1 nodes, after which we give a simple proof of Touchard's formula.