3. Postdoctoral Program - MSRI

This project seeks to use elliptic curves to search for triples (A:B:C) with exceptionally large
q(A:B:C). The approach will be to use properties of certain families of elliptic curves that are
classified according to their isogenous curves. For example, the triple (3 3 :5:2 5 ) has a rather low
quality of q=1.01898, whereas the triple (3:5 3 :2 7 ) has a rather large quantity q=1.42657. These
triples correspond to elliptic curves which are 3-isogeneous to each other.
Project 2: Elliptic Curve Cryptography
There are two aspects when discussing secrets: how to safely encrypt a message so that an
unauthorized party cannot read it and how to effectively decrypt a message so that an authorized
party can. This project sought to have students learn about the Discrete Logarithm Problem by
writing computer code. Students engaged in a friendly competition of passing secrets: one
encrypted, while the other eavesdroped!
The method of encryption using elliptic curves is a type of Public Key cryptosystem, an idea first
put forth by Whitfield Diffie and Martin Hellman in 1976. One takes a message, expresses each
character as a positive integer (using say UTF-8), then encodes the message as a positive integer.
One can then perform mathematics on this integer, using a shared elliptic curve, and then send
this encrypted message to a friend. This process is known as Elliptic Curve Diffie-Hellman
(ECDH).
The method of decrypting using elliptic curves is a variant of Pollard's (p-1) Algorithm, an idea
first put forth by Hendrik Lenstra in 1987. One wishes to solve the Elliptic Curve Discrete
Logarithm Problem (ECDLP) by factoring an integer using elliptic curves. One chooses a
random point on a random elliptic curve and then uses the group law to eventually find a factor
that allows one to invert the discrete logarithms. This process is known as Elliptic Curve
Factorization Method (ECM).
Short Biographies of the 2010 MSRI-UP organizers:
Ivelisse M. Rubio was born and raised in Puerto Rico. She received her B.S. and M.S. in
Mathematics from the University of Puerto Rico-Río Piedras and her Ph.D. in Applied
Mathematics from Cornell University. In 1998, she co-founded the NSF-REU Summer Institute
in Mathematics for Undergraduates (SIMU) at the UPR-Humacao. Ive is currently a Professor in
the Computer Science Department at the UPR-Rio Piedras. Her research interests are finite
fields and applications to error-correcting codes.
Edray Herbert Goins grew up in South Los Angeles, California. A product of the LAUSD
public school system, Dr. Goins attended the California Institute of Technology, where he
majored in mathematics and physics. He earned his doctorate in mathematics from Stanford
University. Dr. Goins is currently an Assistant Professor of Mathematics at Purdue University in
West Lafayette, Indiana. He works in the field of number theory as it pertains to the intersection
of representation theory and algebraic geometry.
Dr. Goins spends most of his summers engaging underrepresented students in research in the
mathematical sciences. He has taught mathematics with the Vanguard Engineering Scholarship
Program through the National Action Council for Minorities in Engineering (NACME), taught
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