Prof. Ashay Dharwadker's Courses (18):

Course	Semester	Grade	Add New Course
Algorithm Design - I	Fall 2000	View	Update Course
Algorithm Design - II	Spring 2001	View	Update Course
Data Structures	Summer 2001	View	Update Course
Probability & Statistics	Fall 2001	View	Update Course
Algorithm Design - I	Fall 2001	View	Update Course
Algorithm Design - II	Spring 2002	View	Update Course
Decision Analysis	Spring 2002	View	Update Course
Probability & Statistics	Summer 2002	View	Update Course
Algorithm Design - I	Fall 2002	View	Update Course
Probability & Statistics	Spring 2003	View	Update Course
Software Engineering	Spring 2003	View	Update Course
Information Systems	Fall 2003	View	Update Course
Database Systems	Fall 2003	View	Update Course
Algorithm Design - II	Spring 2004	View	Update Course
Computer Networks	Spring 2004	View	Update Course
Research	Summer 2005	View	Update Course
Research	Fall 2005	View	Update Course
Research	Spring 2006	View	Update Course

Projects (2)

• Open Visual C++, File - New - Project Name: Browser
• Single Document, Document/View Architecture Support
• None
• None, ActiveX Controls
• Docking Toolbar, Initial Status Bar, Printing and Print Preview, Context Sensitive Help, 3D Controls, Internet Explorer ReBars
• MFC Standard
• Base Class: CHtmlView, Finish
• Build and Execute

f ¹(x) =  f (x)
f ²(x) =  f (  f (x) )
f ³(x) =   f (  f (  f (x) ) )
.
.
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The white points are initially inside a disc of radius 2 centered at the origin and 3 iterations are then performed. Some of the orbits tend to remain bounded while others tend to escape towards infinity.  

Now let X= C ∪ {∞} be the extended complex plane and consider rational functions f (z) = p(z) | q(z) where p, q are polynomials without any common factors. The filled Julia set of the dynamical system ( C ∪ {∞}, p(z) | q(z) ) consists of all points whose orbits are bounded sets in the complex plane C. The Julia set of the dynamical system ( C ∪ {∞}, p(z) | q(z) ) is the boundary of the filled Julia set. A filled Julia set may be a connected set in the complex plane (aka Fatou region) or a totally disconnected set in the complex plane (aka Cantor dust).

We first consider the case f (z) = z ² + c where c is a constant. It is a remarkable fact that the corresponding filled Julia sets are connected (i.e. form Fatou regions) if and only if c belongs to the Mandelbrot set. In fact, this can be taken as a definition of the Mandelbrot set, shown below in Figure 1.

Seminars (1)

• Select prime P=5
• Select prime Q=7
• Find E such that gcd(E,(P-1)*(Q-1))=1 and E%2!=0: Found E=5
• Find D such that ((D*E)%((P-1)*(Q-1)))==1: Found D=5

• Encrypt(T)=pow(T,E)%(P*Q)
• Decrypt(C)=pow(C,D)%(P*Q)
• Your Public Key is (E,PQ). Tell Everybody Your Public Key!
• Your Private Key is D. Keep Your Private Key Secret!

For example, using the ASCII values of characters, the message THE_EAGLE_HAS_LANDED can be encrypted using E and PQ to THJZJAGQJZHAXZQANIJI. This can be decrypted back to the original message only by someone who knows D and PQ. To find your private key D, a hacker needs to factor PQ into P and Q and there are no known methods to do this in a realistic amount of time if PQ is large!

• Explain the above statement. Why does the hacker need to factorize PQ to find your private key?
• Find out how MySQL supports RSA Encryption. Look in the manual!

Research Notes (1)