Download Discrete Mathematics

CORE GROUP
on DISCRETE MATHEMATICS
Prof. Aditya
Shastri
Mr. Sanjay
Sharma
Ms. Somya
Upadhyay
Dr. Reena
Dadhich
Dr. Deepa
Sinha
Mr.Vikas
Pareek
WORK AREAS
1.Combinatorics, Graph theory, Ramsay
theory, Parallel & Distributed Algorithms,
Theory of Computation, E- Commerce & Mobile
Computing
2. Distributed Algorithms
3. Signed Graph Structures
4. Number Theory & Cryptography
5. Signed Graphs in Modeling Communication
6. Graph Coloring & Parallel Algorithms
RESEARCH ACTIVITIES
1. Research Papers, Seminars and Group
Discussions
2. Faculty members are involved in Research
as Ph.D. work, presenting research papers,
attending
conferences,
workshops,
seminars and acquiring expertise in
multiple disciplines.
3. At present three faculty members have
formally registered for Ph.D. in the
Department and many others are working
out their plans.
Prof. Aditya Shastri
Thrust Areas:
– Financial Methods



Complexitity and Approximations of Capacity Auction Variants
Incorporating Business Processes into e-Marketplace
Negotiation Mechanisms
Auctions of Digital / Intellectual Goods
– Bio Metrics : Fingerprint Identification
– Design and Analysis of Communication Networks
– Ramsey Theory
– Classical Graph Theory and Combinatorics
Dr. Reena Dadhich (Sr. Lecturer)
Her doctoral research work is based on Analysis and
Design of Wireless ad-hoc networks with low
Forwarding Index. For efficient communication
between nodes, ad-hoc networks are typically
grouped into clusters, where each cluster has a
cluster head (or Master). Cluster head nodes are also
responsible for forwarding of messages in to the
clusters. Consequently, the cluster head tend to
become potential points of failures and naturally,
there occurs congestion. (Because large no. of routes
passes through the cluster head node.)
Therefore, it is important to control the congestion
using some efficient routing algorithms. It can be
achieved by evenly distributing the routing and
load among all nodes in the network, i.e. by
assigning the responsibilities of being (Master)
cluster head node to some other under loaded
node, who has enough capability, i.e. load
balancing. This under loaded node can be a:
1.) Master node or Slave node in another Cluster
over the whole network .
or
2.) Can be a Slave node in the same cluster in
which the Master node is over loaded
List of Publications
1. R. Dadhich “Forwarding Index And Wireless Ad hoc Networks”, Proceedings of
a National Seminar on Communication Networks, IIT Roorkee, INDIA, Dec.
2003.
2. R. Dadhich “Design of Wireless Ad hoc Networks with Weighted Clustering and
Low Forwarding Index”, Proceedings of ITWINS held at Thapar Institute of
Tech., Patiala, INDIA, Dec. 2004.
3. R. Dadhich, “Load Balancing in Wireless Adhoc Networks”, Proceedings of
National Conference on Computing –2005, CSI Indore Chapter, INDIA, May
2005.
4. A. Shastri and R. Dadhich , “Load Balancing in Wireless Ad Hoc Networks with
Low Forwarding Index”, Proceedings of IEEE International Conference on
Information and Automation, held at COLOMBO, Sri Lanka during Dec. 2005.
5.
A. Shastri and R. Dadhich , “Load Balanced Routing in Wireless Ad-hoc
Networks” Proceedings of 3rd International Conference ObCom-2006: Mobile,
Ubiquitous & Pervasive Computing, December 16-19, 2006 VIT, Vellore, TN,
INDIA.
Dr. Deepa Sinha, Sr. Lecturer
Her work is on discrete structures called signed
graphs which are essentially networks. A typical
signed graph is shown below:
S3
S1
S2
S4
S5
S6
A signed
Digraph
In such a signed networks their networks arcs are being
designated as being positive or negative depending on the
nature of interaction that is categorized as being
supportive or inhibitive in some given sense specified in a
given context. In real Life to study the Dynamics of any
system it is necessary to know the interaction pattern
amongst the submodules of the system with the
description of how precisely one is able to represent the
positive and negative aspects of various links
interconnecting the submodules. In most of the
engineering
and technological systems, a proper
understanding of such networks called generally the
structural (modular) configuration of the systems is
essential in their proper operation by means of taking
care of various risk factors, optimal operating conditions,
maintenance etc.
The work carried out by her on the signed graphs as such
mainly deals with the structural reconfigurations of the
dynamical systems under prescribed rules and rules are
designed to deal with a variety of interconnections
amongst the elements of the system. Equivalence or
stability of structures under such transformations are
required to be known under such prescribed rules and one
needs to study the conditions under which equivalence and
stability occurs. We have thus taken up research initiative
on studying such equivalence of structures under such
transformation.
LIST OF PUBLICATIONS
Mukti Acharya and Deepa Sinha, A characterization of signed graphs that are
switching equivalent to their jump sigraphs, Graph Theory Notes, New York
Academy of Sciences, New York, XLIII (2002), 7-8.
Mukti Acharya and Deepa Sinha, A characterization of sigraphs whose line
sigraphs and jump sigraphs are switching equivalent, Graph Theory Notes of
New York Academy of Sciences, New York, XLIV (2003), 30-34.
Mukti Acharya and Deepa Sinha, A characterization of line sigraphs, Extended
Abstract In: Electronic Notes in Discrete Mathematics, 15 (2003).
Mukti Acharya and Deepa Sinha, characterizations of line sigraphs, Nat. Acad.
Sci.-Letters, 28(1-2)(2005), 31-34.
Mukti Acharya and Deepa Sinha, Common Edge Sigraphs, AKCE
International Journal of Graphs and Combinatorics, 3, 2(2006), 155-130.
Publications of Book
AMIETE Question Answer series, twelve (six + six) solutions to six
question papers of Mathematics and six question papers of Numerical
analysis & Computer Programming-(1998-2001)
Mr. Vikas Pareek
INTEGER FACTORIZATION AND CRYPTANALYTIC
ATTACKS ON RSA
Integer factorization is one of the most challenging problems of
number theory and hardness of this problem is behind the security
of RSA cryptosystem.
The proposed work aims to find the possibility of a fast
factorization algorithm. It presents geometrical interpretation of
factorization problem (a restricted version for RSA modulus). Then
the proposed algorithm is compared with the Fermat’s method of
factorization (Fermat’s factorization method fails to work
efficiently if the prime factors of the number are far apart).
Different cryptanalytic attacks on RSA will also be explored.
- guided projects on implementation of RSA and ftp client using
chaotic function cryptography and some web based projects.
Ms. Somya Upadhayay, Lecturer
Somya Upadhayay will be working on the
application of Theory of locally interacting and
product potential networks of automata to
modeling balance in Social groups and similarly
studying the Signed graph structures in
communication.
Mr. Pravin Garg, Lecturer
He is working in the field of Boundary layer
theory and is putting forward to have a discrete
analysis of same.
Mr. Sanjay Sharma, Lecturer
He will be working on the application part of signed
graph equations and its authentic algorithms design.
He has guided the projects on implementation of
parallel-algorithms using the shared memory, e-mail
client network application.
He is now working on comparative study of time
complexity of various parallel algorithms (dataparallelism) using multiprocessors computer and
linear algorithms.
In
M.Sc.
(Mathematical
Sciences),
Pure
Mathematics it is in curriculum to review one of
the research paper and write small research
paper
in
their
last
semester.
Few representative research papers:
1.
Signed Graph Portfolio in risk management
2.
Negation Switching invariant 1-path sigraphs
3.
Connectivity and Transformation Graphs
4.
A Characterization of Sigraphs whose Line Sigraphs
and Jump Sigraphs are Switching Equivalent
5.
A Family of Minimally Circular Imperfect Graphs
6.
Minimum cost homomorphism problem for directed
and undirected Graphs
7.
3-Sigraphs
8.
List Coloring and the number of colorings of the
graph
Proposal to set a Crypto Research Cell
As number theory (and esp. its applications to
Cryptography) is one of the thrust areas specified
by DST, Govt. of India, a research cell for
Cryptography can be constituted under the
auspices of CMS. Presently leading centers of
research like IISc Banagalore, ISI Kolkata and
Microsoft’s R&D lab are working on
cryptography.
 It will help us utilize the mathematical skills of
our faculty members and students.
The cell may include following:
Theoretical research
•Elementary Number theory, Finite fields,
Arithmetic and algebraic algorithms, secret key
and public key cryptosystems, Block and stream
ciphers, Probabilistic encryption, Elliptic curves,
Hard functions
•Number theoretic problems viz., Integer
factorization, Catalan’s Conjecture, Twin-primes
conjecture, Mersenne numbers, algorithmic
aspects of cryptosystems, etc.
Laboratory work
•Random number generators, Probabilistic analysis,
randomized algorithms, embedded systems security
•S/W: Cryptanalytic
cryptosystems
attacks
on
public
key
Performance evaluation of cryptographic algorithms
Cryptography with chaotic functions.
Teganography applications for video files.
H/W: Hardware implementation of trapdoor ciphers,
etc.
The focus will be on algorithmic aspects of number theory
and building applications for cryptography and related
areas.
This cell will include people from Computer Science,
Mathematics, Statistics, Physics and Electronics, so
it will provide a good platform for inter-disciplinary
research and will promote synergy in them.
Faculty members:
1. Dr. Vinod Patidar (Dept. of Physics)
2. Mr. Vikas Pareek (Dept. of Computer Science)
3. Mr. Pravin Garg (Dept. of Mathematics)