### PROJECT

### CRYPTOGRAPHY

**TÜBİTAK Integrated PhD Program: Homogenous spaces, enumerative geometry and singularity theory**

Collaborators

Ersan Akyıldız, Department of Mathematics & Institute of Applied Mathematics, METU (Coordinator)

F. Arslan, Department of Mathematics, METU

J. B. Carrell, University British Columbia

Y. Civan, Süleyman Demirel University

Ö. Kişisel, Department of Mathematics, METU

H. Önsiper, Department of Mathematics, METU

P. Pragacz, Polish Academy of Sciences

M. Tosun, Galatasaray University

Funded by TÜBİTAK

### FINANCIAL MATHEMATICS

**ESF(European Science Foundation) Research Networking Project: Advanced Mathematical Methods for Finance**

Collaborators

H. Körezlioğlu (Coordinator), Ö. Uğur, A. Hayfavi, K. Yıldırak, Institute of Applied Mathematics, METU

E. Gaygısız, Department of Economics & Institute of Applied Mathematics, METU

B. Karasözen, Department of Mathematics & Institute of Applied Mathematics, METU

Funded by ESF, TÜBİTAK, Scientific and Technical Research Council of Turkey

### SCIENTIFIC COMPUTING

- DAAD
**TÜBİTAK Integrated PhD Program: Continuous Optimization Methods and Applications**

Collaborators

B. Karasözen, Department of Mathematics & Institute of Applied Mathematics, METU (Coordinator)

G.W. Weber, Institute of Applied Mathematics, METU

T. Ergenç, Department of Mathematics, Atilim University

Y. Uludağ, Department of Chemical Engineering & Institute of Applied Mathematics, METU

O. Stein, Institute for Operations Research, University of Karlsruhe, Germany

G. Still, Faculty of Mathematics, University of Twente, The Netherlands

A. Rubinov, A. Bagirov, School of Information Technology & Mathematical Sciences, University of Ballarat, Australia

P. Toint, Department of Mathematics, The University of Namur, Belgium

S. Tijs, CentER and Department of Econometrics and Operation Research, Tilburg University, The Netherlands

Z. Volkovich, Software Engineering Department, ORT Braude College, Israel

R.H.W. Hoppe, Department of Mathematics, University of Houston & University of Augsburg, USA, Germany

**TÜBİTAK-NSF INT Project: Development of Modelling and Optimiztion Tools for Hybrid Systems**

We consider the modeling and optimization of dynamic hybrid systems. Such systems occur widely in nature (particular in biological, chemical and mechanical applications) and are also essential components of economic planning models. For optimization formulations, these discontinuous decisions are represented as binary variables, leading to mixed integer nonlinear programs (MINLPs), or through complementarity relations, leading to mathematical programs with complementarity constraints (MPCC). Our research will consider both MINLP and MPCC problems over a broad set of application domains and we intend to develop new optimization strategies that combine MPCC and MINLP algorithms within a single strategy. These research results will be applied to broad set of applications including systems biology and drug design, planning and scheduling models for enterprise wide optimization, applications of hybrid models in mechanical and process engineering systems. Finally, these results of this research will allow consideration of challenging dynamic hybrid systems too large to be considered with existing optimization strategies.

Collaborators

L. Biegler, University of Carnegie Mellon, Department of Chemical Engineering (Coordinator)

B. Karasözen, Department of Mathematics & Institute of Applied Mathematics, METU (Coordinator)

M. Türkay, Koç University, Departemnt of Industrial Engineering

H. Oktem, Institute of Applied Mathematics, METU

A. Tezel, Department of Mathematics, METU

Funded by TUBITAK-NSF 2005-2008, Scientific and Technical Research Council of Turkey

** **

**TÜBİTAK-Career Project: Modeling multistationary processes by using hybrid system formulation: A study with priority on functional genomics**

We consider inferring regulatory relations, in particular gene interactions from the empirical measurements, in particular expression microaarray profiles. Picewise linear formulations are studied as abstract models but they do not include an intervention model. In this context the task is inferring the dynamics of each stationary steady state by using a piecewise analytically computable model, while possible switchings between states are detected by statistical methods. Considering delays in the models of nonlinear dynamical systems create problems in the accurate computation of the models. Thus, most of the research on delays considers the cases where the effects of delays can be simplified. In this context this projects aims at developing model classes evolving in continuous time without ignoring or simplifying the effects of delays.

Coordinator

H. Öktem, Institute of Applied Mathematics, METU

Funded by TUBITAK-104T133, Scientific and Technical Research Council of Turkey

** **

**TÜBİTAK-Research Project: The Boundary Element Solutions of The Nuclear Fusion Reactor Problems**

The magnetohydrodynamic equilibrium in an axisymmetric plasma is described by the Grad-Shafranov (GS) equation in terms of the magnetic flux. Due to the non-linear nature of the Grad-Shafranov equation a general analytical solution is not possible. However, for a given current density (the non-homogeneouity), the Grad-Shafranov equation can be solved numerically. In this project two different recent and powerful numerical methods, namely the finite element method with residual free buble functions and the boundary element method are going to be used without the need of iteration. The residual free bubble functions are used for stabilizing the numerical results obtained from the finite element method. The boundary element method is well-suited for plasma equilibrium analysis that requires efficient data preparation and computation following the change in plasma shape during the operation of an actual fusion device. The nonhomogenouity in the Grad-Shafranov equation is going to be approximated by using radial basis functions.

The GS equation is the the axisymmetric case of the MHD equations, in terms of flux function with some simplifications . Therefore, the general case of MHD equations will be solved numerically. The two-dimensional MHD equations are defined in terms of momentum equations(Navier-Stokes equations), Maxwell equation and continuity equation. Firstly, the Navier-Stokes equations are solved in primitive variables form with finite element method. Oscillations coming from the solution of the coupled form are been solved by using stabilized methods. Finally MHD terms are added and full form of the equations are solved with two level finite element method using residual free bubble functions.

Collaborators

M.Tezer-Sezgin, Department of Mathematics & Institute of Applied Mathematics, METU (Coordinator)

A.I. Neslitürk, Department of Mathematics, Izmir Institute of Technology & Institute of Applied Mathematics, METU

S. Han Aydin, Institute of Applied Mathematics, METU

S. Gümgüm, Institute of Applied Mathematics, METU

Funded by TUBITAK-105T091, Scientific and Technical Research Council of Turkey

## Participating Projects

### CRYPTOGRAPHY

### FINANCIAL MATHEMATICS

### SCIENTIFIC COMPUTING

** **

**Usage of Datamining techniques in Production Quality Assesment**

Colloborators

G. Koksal, Department of Industrial Engineering, METU (Coordinator)

B. Karasozen, Department of Mathematics & Institute of Applied Mathematics, METU

G.W. Weber, B. Akteke, S.Ozogur, Institute of Applied Mathematics, METU

Funded by TUBITAK 2006-2008, Scientific and Technical Research Council of Turkey

** **

**Meteorology and Oceanography Network of Excellence MONOE**

Colloborators

E. Ozsoy , Department of Marine Sciences, METU (Coordinator)

B. Karasozen, Department of Mathematics & Institute of Applied Mathematics, METU

H. Oktem, O. Ugur, Institute of Applied Mathematics, METU

Funded by TUBITAK 2006-2008, Scientific and Technical Research Council of Turkey

## Past/Completed Projects

### CRYPTOGRAPHY

** **

**Research and Development in Cryptography : Cryptographic Algorithm Design, Analysis and Implementation**

The project aims at advancing the state of the art in the field of cryptography by doing research in areas like developments of new methods for the algorithm design, analysis and implementation as well as establishing the required infrastructure in the institude for cryptographic research.

Collaborators

Ersan Akyıldız, Department of Mathematics & Institute of Applied Mathematics, METU (Coordinator)

Rüyal Ergül, Melek Yücel, Department of Electrical Engineering & Institute of Applied Mathematics, METU

Ali Doganaksoy, Ferruh Özbudak, Muhiddin Uğuz, Emrah Çakçak, Department of Mathematics & Institute of Applied Mathematics, METU

Funded by DPT, State Planning Organization

** **

**Research, Development and Implementations about Public Key Infrastructures**

This project intends to follow up the latest developments on Digital Signature algorithms (RSA, DSA and ECDSA) and develop software tools to test their security. A reference system will be set up to investigate implementation issues in signature generation and distribution.

Collaborators

Rüyal Ergül, Department of Electrical Engineering & Institute of Applied Mathematics, METU (Coordinator)

Ersan Akyıldız, Ali Doğanaksoy, Ferruh Özbudak, Muhiddin Uğuz, Department of Mathematics & Institute of Applied Mathematics, METU

Mustafa Alkan, K.Sacit Sarıkaya, Sezen Yeşil, Özgür Öztürk, Onur Gençer, Telecommunication Authority

Funded by TÜBİTAK, Scientific and Technical Research Council of Turkey

** **

**Selecting Secure Elliptic Curves GF(p) and Implementing a Signature System Based on Elliptic Curves**

This project aims to develop software tools to generate secure elliptic curve families in GF(p) fields and make a selection among suitable candidates that satisfies a set of requirements. An imlementation has been planned to investigate problems associated with signature systems based on elliptic curve cryptography.

Collaborators

Ersan Akyıldız, Department of Mathematics & Institute of Applied Mathematics, METU(Coordinator)

Rüyal Ergül, Department of Electrical Engineering & Institute of Applied Mathematics, METU (Coordinator)

Funded by ASELSAN A.Ş.

**Developing Statistical and Structural Test Suite Software to Evaluate the Security of Block Ciphers**

Tests that are statistical and structural capable of analyzing any practical block cipher are the subject of this project. Statistical and Structural test suite software will be used to evaluate the security of block ciphers.

Collaborators

Ali Doğanaksoy, Department of Mathematics & Institute of Applied Mathematics, METU(Coordinator)

Funded by ASELSAN A.Ş.

**Examining, Improving and Implementation of Error-Correcting Code Generation Methods**

The aim of the project is to construct new authentication codes having good parameters and improve the parameters of some known codes. In general,some techniques from finite fields and Galois rings are used for the constructions.

Collaborators

Ferruh Özbudak, Department of Mathematics & Institute of Applied Mathematics, METU(Coordinator)

Murat Cenk, Emrah Çakçak, Zülfükar Saygı Institute of Applied Mathematics, METU(Researchers)

Funded by BAP, Scientific Research Support

### FINANCIAL MATHEMATICS

**DAAD Program Subject-Related Partnerships with Universities in Developing Countries: Financial and Insurance Mathematics**

Collaborators

R. Korn, Department of Mathematics & Frauenhofer Geseallschaft, University of Kaiserslauen(Coordinator)

J. Lehn, R. Fischer, M. Grombach, Department of Mathematics, University of Technology, Darmstadt

H. Körezlioğlu, O. Ugur, A. Hayfavi, K. Yildirak, Institute of Applied Mathematics, METU

E. Gaygısız, Department of Economics & Institute of Applied Mathematics, METU

B. Karasözen, Department of Mathematics & Institute of Applied Mathematics, METU

Funded by University of Kaiserslautern and DAAD(German Academic Exchange Service) 2003-2006

### SCIENTIFIC COMPUTING

**Optimization of Stirrer Configuration in a Tank**

Mixing of fluids in stirred tanks, being one of the oldest of unit operations, is commonly used by chemical, biotechnological, harmaceutical, food processing etc. industries. In the view of the fact that the geometrical configuration of the tank and impeller type varies depending on the kind of operation that is going to be carried out, it is an awkward task to select the most suitable tank for the process to reach the desired product quality. The main objective of the project is to develop and apply a numerical optimization tool to determine the optimum configuration parameters of stirrer configurations.

Collaborators

M. Schaefer, Department of Mechanical Engineering, University of Technology Darmstadt(Coordinator)

B. Karasözen, Department of Mathematics & Institute of Applied Mathematics, METU (Coordinator)

Yusuf Uludağ, Department of Chemical Engineering & Institute of Applied Mathematics, METU

G. Karakaş, K. Yapıcı, Department of Chemical Engineering, METU

H. Aksel, Department of Mechanical Engineering & Institute of Applied Mathematics

O. Uğur, Institute of Applied Mathematics, METU

Funded by Volkswagen Foundation, 2002-2005