CORE GROUP on DISCRETE MATHEMATICS

1. CORE GROUP on DISCRETE MATHEMATICS Prof. Aditya Shastri Mr. Sanjay Sharma Ms. Somya Upadhyay Dr. Reena Dadhich Dr. Deepa Sinha Mr.Vikas Pareek

2. 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

3. RESEARCH ACTIVITIES • Research Papers, Seminars and Group Discussions • Faculty members are involved in Research as Ph.D. work, presenting research papers, attending conferences, workshops, seminars and acquiring expertise in multiple disciplines. • At present three faculty members have formally registered for Ph.D. in the Department and many others are working out their plans.

4. 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

5. 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.)

6. 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

7. 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. • 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.

8. 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 S6 S4 S5 A signed Digraph

9. 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 preciselyone 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.

10. 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 studyingsuchequivalence of structures under such transformation.

11. 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)

12. 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.

13. 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.

14. Mr. Sanjay Sharma, Lecturer Hewill 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 (data-parallelism) using multiprocessors computer and linear algorithms.

15. 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: Signed Graph Portfolio in risk management Negation Switching invariant 1-path sigraphs Connectivity and Transformation Graphs ACharacterization of Sigraphs whose Line Sigraphs and Jump Sigraphs are Switching Equivalent A Family of Minimally Circular Imperfect Graphs Minimum cost homomorphism problem for directed and undirected Graphs 3-Sigraphs List Coloring and the number of colorings of the graph

16. 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.

17. 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.

18. Laboratory work • Random number generators, Probabilistic analysis, randomized algorithms, embedded systems security • S/W: Cryptanalytic attacks on public key cryptosystems • Performance evaluation of cryptographic algorithms • Cryptography with chaotic functions. • Teganography applications for video files. • H/W: Hardware implementation of trapdoor ciphers, etc.

19. 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: • Dr. Vinod Patidar (Dept. of Physics) • Mr. Vikas Pareek (Dept. of Computer Science) • Mr. Pravin Garg (Dept. of Mathematics)