Mathematica and the Science of Secrecy
In January of 1999, Sarah Flannery was named the Irish
Young Scientist of the Year for her work in devising a highly innovative,
fast, and secure new algorithm for data encryption of internet and other
electronic communications. The award carried with it an impressive silver
trophy, a monetary sum of IR£1,000, a oneweek trip to Thessaloniki,
Greece, to represent Ireland at the European Contest for Young Scientists,
and a wave of international media attention. It is largely due to
all of this interest that Sarah was asked to write about her
experiences. The resulting book, In Code: A Mathematical Journey,
was published in April 2000.
Sarah's project, entitled "CryptographyA New Algorithm versus the RSA,"
was widely praised by both competition judges and professional
cryptographers alike for its brilliant applications of number theory and
its demonstration of a strong grasp of the fundamentals of cryptography.
In it, Sarah compared the algorithm she developed, called the
CayleyPurser (CP) algorithm, to the RSA public key cryptosystem developed
in the 1970s at MIT. Her intent was to demonstrate that the CP algorithm
was much faster, and just as secure, as the celebrated RSA. Along the way,
however, she also discovered how "wonderfully powerful" Mathematica
is. It was "great fun learning the Mathematica commands to
[encipher and decipher]...just seeing it work was a real kick," she wrote.
Sarah initially undertook to explore the science of cryptography in the
fall of 1997 during her Transition year, after being introduced to the
topic in a class called "Mathematical Excursions" taught by her father at
the Cork Institute of Technology. When her science teacher, who was
looking for competition participants, described the nature of the fun,
excitement, and experiences at the nationally known Esat Young Scientist &
Technology Exhibition, Sarah "knew it would be worth doing something just
for the craic to be had up in Dublin for five days." They were currently
discussing cryptography in her father's class. As she
had "heard Dad rave about this mathematical package called
Mathematica that he made out to be the cat's pyjamas,"
Sarah decided that this would
be an ideal opportunity to learn what this "wonderful piece of software"
could do. Besides, she "had some inkling that [she] would really love
programming."
In her book, Sarah includes a lively, comprehensive discussion of the
fundamentals of cryptography and the underlying mathematics and number
theory behind it. She gradually pulls the reader in with early
presentations and explanations of some popular and some notsocommon
logic and math puzzles. Readers are invited to either think their way
through the introductory chapters or bypass them for a still very
captivating description of her cryptography project and the challenges
involved, which were many. Sarah's project required that she learn number
theory, the necessary cryptographic ideas, and programming in
Mathematica so as to implement and illustrate the cryptographic
schemes, all within a few months. As she learned additional number theory and
the way to program the code to manipulate it, she was thrilled by how
Mathematica processed "in the blink of an eye" the huge numbers common in
cryptography.
Sarah entered her first project, "CryptographyThe Science of Secrecy," in
the 1998 Esat competition, which earned her the Intel Excellence Award,
amongst other honors. It also helped her to land a valuable work
experience with Baltimore Technologies, a Dublinbased cryptography
company, where she was assigned to investigate an idea for a possible new
public key cryptosystem presented in an unpublished paper written by the
company's founder, Michael Purser. It was following this exposure that
Sarah realized she might be able to extrapolate from Purser's ideas to
devise an asymmetric algorithm that exploited the noncommutative property
of matrix multiplication. Whereas the RSA algorithm
requires modular exponentiation of numbers from two hundred to five hundred digits in
length to numbers of the same magnitude, this new algorithm would involve
only modular multiplication, making it much faster.
Over the next few months, Sarah set herself to the task of learning basic
matrix theory in order to construct her new algorithm, writing programs to
generate examples of unfamiliar concepts and consulting her father and
numerous journals for additional information. As her mathematics knowledge
and programming skills increased, she determined to write cleaner, more
sophisticated code after admiring the work of others that demonstrated
the "wonderful flexibility of the Mathematica language." Using
Mathematica, Sarah implemented both the RSA and the CP algorithms and
performed comparative runtime analyses on them. Her efforts were rewarded
when she was able to demonstrate successfully that the CP algorithm was
20 to 30 times faster than the RSA and to knowledgeably defend an attack on it
as well. This all garnered her a first prize in the European Young
Scientist Competition and a trip to attend the December 1999 Nobel Prize
ceremonies in Stockholm.
Apart from her extraordinary accomplishments, Sarah considers herself to
be a fairly regular girl with a wide range of hobbies and interests. She
has recently completed her Irish Leaving Certificate Examinations, and
although she is considering taking a year off to independently study more
mathematics, she will eventually go on to a university, where she will study
mathematics and computing. "I have no doubt that I am not a genius," says
Sarah. However, she was fortunate to find a project topic in which she was
"interestedvery interestedand worked very hard." She was also fortunate
enough to have her father as a mentor. He always encouraged her to learn
the fundamentals, "just to think," and not to be intimidated by unfamiliar
problems or topics.
An indepth mathematical description and comparison of the RSA and CP
algorithms, including the Mathematica code, is available at
http://www.cayleypurser.ie. Sarah's story, In Code: A
Mathematical Journey, is currently for sale throughout the UK, with
plans for translation and distribution in Japan, Korea, China, and the
United States within the year. It has so far received only excellent
reviews and is already in its second printing. It can be ordered online
through the Amazon.co.uk bookstore.

