HRUMC XIV a Success!

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Department of Mathematics April 27, 2007

Last weekend, about 20 Union students and faculty members attended the 14th annual Hudson River Undergraduate Mathematics Conference. Approximately 500 people from colleges and universities throughout the northeast attended the conference, and more than 175 talks were delivered, including ten by Union College folk: Susan Beckhardt, Laura Hutchinson, Professor Brenda Johnson, Tom Mazur, Jacqueline Palermo, Jaclyn Siedlecki, Daniel Stevenson, Sydnie Wells, Peter Wright, and Leah Ziamandanis. Great job! Additional kudos to Professors Brenda Johnson and Bill Zwicker, members of the conference steering committee, for doing a wonderful job helping to plan and organize this huge affair. On the horizon: In two years, Union College will (likely) be hosting HRUMC XVI. Students at the host institution play an invaluable role in planning and organizing the conference. So, our current first- and second-year students will have the fantastic opportunity to get involved in this process!

Above: From left, Sarah Jensen, Dan Stevenson, Professor Julius Barbanel, Susan Beckhardt, and Laura Hutchinson. Below: Susan Beckhardt giving her talk.

Above: From left, Jackie Palermo, Sydnie Wells, Peter Wright, Jackie Siedlecki, and Leah Ziamandanis. Below: Professor Johnson and Tom Mazur before their talks.

When I was a child my stepmother went back to college to get her Ph.D. She complained incessantly about this thing called a “thesis.” Whatever this monster was, I knew that it would be a very long time before I had to find out. Well as a senior in my final term of college I feel really old now that my thesis has come and gone. I have to say, the time I spent writing my thesis flew by. I spent three years as a Math major without a clue as to what in particular interested me. Finally, spring term of last year, I took Cryptology with Professor Kathryn Lesh. I knew immediately that I wanted to write a Crypto thesis. The bond that I formed with my thesis advisor over the past few months is a most unique relationship. Not only have we come to respect each other, but we’ve also developed a great friendship fostered by our ability to make each other laugh. There were so many times when I was so frustrated with my work, but she was always equally as patient and encouraging in return. Choosing an article to research was a tedious process because it was difficult to estimate whether or not a paper will be manageable. I chose an article called, “Public-key Encryption Based on Chebyshev polynomials,” by L. Kocarev et.al. Public-key encryption is used to transfer secret information over insecure lines without prior communication over a secure line, and Chebyshev polynomials are a class of polynomials defined recursively. I found that public-key cryptography is a field that is

always evolving. Cryptographers are working on new cryptosystems that are more secure and efficient than previously existing ones. In the search for such algorithms, the use of Chebyshev polynomials has been proposed. In my thesis, I describe modified versions of two well know cryptosystems, RSA and El-Gamal, that replace the use of exponentiation with the use of Chebyshev polynomials. I show that the modified systems are as practical and secure as the classical systems. In the classical cryptosystems I learned about last spring, a cipher text was calculated by exponentiating the message, represented as an integer, to the degree of the published encryption key. To decrypt, the cipher text was exponentiated to the degree of the corresponding decryption key. I studied a modified version of these algorithms that encrypt the message by finding the Chebyshev polynomial of the degree of the published encryption key evaluated at the value of the message. To decrypt such a cipher text, one finds the Chebyshev polynomial of the degree of the secret decryption key evaluated at the value of the cipher text. I recently presented a portion of my work at the HRUMC, which was a truly rewarding process. I’m actually still in shock and find it hard to believe that I wrote a senior thesis. While writing my thesis, I realized how much I enjoy not only researching Math, but also teaching it. I’ve since decided to pursue my doctorate degree in Math. I’ll never forget the experience I had last term or the relationships I have built within the Math department.

Problem of the Newsletter: April 27, 2007 Congratulations to Brandon Bartell ’10 and Schuyler Smith for submitting a correct solution to last week’s problem of the newsletter. You can view a winning solution on the bulletin boards around Bailey Hall.

Here is this week’s problem: Find integers

!

x and

!

y such that

!

18+ 260 = x + y . Professor Friedman will accept solutions to this problem until 12:00 noon Thursday, May 3.

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Pieces from Theses: A View from Jackie Palermo (’07)

Session II: 10:30 AM to 12:10 PM, in Olin 107: • Philip Biller: “Using Polynomial and Normal Bases

to Represent Finite Fields” • Jessica DiMarco: “John L. Drost’s Proof of

Ramanujan’s Mod 7 Congruence Theorem” • Thomas Mazur: “Homology and Detector Networks” • David Osei-Amoateng: “Infinite Series and

Approximations of Pi“ • Jacqueline Palermo: “Public-key Encryption With

Chebyshev Polynomials”

Session V: 3:30 to 5:10 PM, in Olin 107: • Susan Beckhardt: “Graphs with Knotted

Hamiltonian Cycles” • Giselle Parrelli: “Germain’s Theorem” • Jaclyn Siedlecki: “Incommensurable Magnitudes,

Irrational Numbers, and the Pythagoreans” • Peter Wright: “Relating Functions to Relations” • Leah Ziamandanis: “Ancient Greek Mathematics: A

look at the first notions of calculus”

Ten Math Talks at Steinmetz Symposium, Friday May 4, 2007