Mathematics and Computer Science: The Interplay

Mathematics has been an important intellectual preoccupation of man for a long time. Computer science as a formal discipline is about seven decades young. However, one thing in common between all users and producers of mathematical thought is the almost involuntary use of computing. In this article, we bring to fore the many close connections and parallels between the two sciences of mathematics and computing. We show that, unlike in the other branches of human inquiry where mathematics is merely utilized or applied, computer science also returns additional value to mathematics by introducing certain new computational paradigms and methodologies and also by posing new foundational questions. We emphasize the strong interplay and interactions by looking at some exciting contemporary results from number theory and combinatorial mathematics and algorithms of computer science

Recent Trends in Applied Cryptology

The science of secret communication namely cryptology is an important area of research. It has come up as a multidisciplinary subject including Mathematics, Electronics, Communication and Computer Science. With advancements in computers and communication technologies, many systems have turned digital. The need for protection of stored and transmitted information by corporate and government agencies has brought this subject Into the public domain in addition to the traditional domains of military and government sectors. In the last three decades, the technical developments in this area have grown manifolds. In this paper we give an account of developments in this field. Including a brief history, some recent advances, certain important applications and a few future trends

Stochastic Differential Equations for Linear Smoothing Problems

Stochastic differential equations for the linear fixed point, fixed interval, and fixed lag smoothing problems are derived using the martingale representation theory

An approximate algorithm for the minimal vertex nested polygon problem

Given two simple polygons, the Minimal Vertex Nested Polygon Problem is one of finding a polygon nested between the given polygons having the minimum number of vertices. In this paper, we suggest efficient approximate algorithms for interesting special cases of the above using the shortest-path finding graph algorithms

On the cubic sieve method for computing discrete logarithms over prime fields

In this paper, we report efficient implementations of the linear sieve and the cubic sieve methods for computing discrete logarithms over prime fields. We demonstrate through empirical performance measures that for a special class of primes the cubic sieve method runs about two times faster than the linear sieve method even in cases of small prime fields of the size about 150 bits. We also provide a heuristic estimate of the number of solutions of the congruence $X^{3}?=?Y^{2}Z$ (mod p) that is of central importance in the cubic sieve method

Stack and queue number of 2-trees

We consider the two problems of embedding graphs in a minimum number of pages and ordering the vertices of graphs in the form of queue layouts. We show that the class of 2-trees requires 2-pages for a book embedding and 3-queues for a queue layout. The first result is new and the latter result extends known results on subclasses of planar graph

Image Encryption with Space-filling Curves

Conventional encryption techniques are usually applicable for text data and often unsuited for encrypting multimedia objects for two reasons. Firstly, the huge sizes associated with multimedia objects make conventional encryption computationally costly. Secondly, multimedia objects come with massive redundancies which are useful in avoiding encryption of the objects in their entirety. Hence a class of encryption techniques devoted to encrypting multimedia objects like images have been developed. These techniques make use of the fact that the data comprising multimedia objects like images could in general be seggregated into two disjoint components, namely salient and non-salient. While the former component contributes to the perceptual quality of the object, the latter only adds minor details to it. In the context of images, the salient component is often much smaller in size than the non-salient component. Encryption effort is considerably reduced if only the salient component is encrypted while leaving the other component unencrypted. A key challenge is to find means to achieve a desirable seggregation so that the unencrypted component does not reveal any information about the object itself. In this study, an image encryption approach that uses fractal structures known as space-filling curves- in order to reduce the encryption overload is presented. In addition, the approach also enables a high quality lossy compression of images

Performance Comparison of Linear Sieve and Cubic Sieve Algorithms for Discrete Logarithms over Prime Fields

It is of interest in cryptographic applications to obtain practical performance improvements for the discrete logarithm problem over prime fields F-p with p of size less than or equal to 500 bits. The linear sieve and the cubic sieve methods described in Coppersmith, Odlyzko and Schroeppels paper [3] are two practical algorithms for computing discrete logarithms over prime fields. The cubic sieve algorithm is asymptotically faster than the linear sieve algorithm. We discuss an efficient implementation of the cubic sieve algorithm incorporating two heuristic principles. We demonstrate through empirical performance measures that for a special class of primes the cubic sieve method runs about two to three times faster than the linear sieve method even in cases of small prime fields of size about 150 bits

On Assigning Prefix Free Codes to the Vertices of a Graph

For a graph G on n vertices, with positive integer weights $w_1, . . . , w_n$ assigned to the n vertices such that, for every clique K of G, \[ \sum_{i \in K}{\frac{1}{2^{w_i}}} \leq 1 \], the problem we are interested in is to assign binary codes $C_1, . . . , C_n$ to the vertices such that $C_i$ has $w_i$ (or a function of $w_i$) bits in it and, for every edge \{i, j\}, $C_i$ and $C_j$ are not prefixes of each other.We call this the Graph Prefix Free Code Assignment Problem. We relate this new problem to the problem of designing adversaries for comparison based sorting algorithms. We show that the decision version of this problem is as hard as graph colouring and then present results on the existence of these codes for prefect graphs and its subclasses

Efficient Dictionary for Salted Password Analysis

User authentication is essential for accessing computing resources, network resources, email accounts, online portals etc. To authenticate a user, system stores user credentials (user id and password pair) in system. It has been an interested field problem to discover user password from a system and similarly protecting them against any such possible attack. In this work we show that passwords are still vulnerable to hash chain based and efficient dictionary attacks. Human generated passwords use some identifiable patterns. We have analysed a sample of 19 million passwords, of different lengths, available online and studied the distribution of the symbols in the password strings. We show that the distribution of symbols in user passwords is affected by the native language of the user. From symbol distributions we can build smart and efficient dictionaries, which are smaller in size and their coverage of plausible passwords from Key-space is large. These smart dictionaries make dictionary based attacks practical