While many instructors are now using cryptography to “spice up” their
number theory courses, most stick either to classical ciphers like the
Caesar cipher or to the RSA cryptosystem. While these are indeed good
examples of the use of number theory in cryptography, instructors may
not be aware that the new Advanced Encryption Standard (AES) also uses
quite a bit of number theory, in the guise of finite field arithmetic,
or modular arithmetic of polynomials. While the generalization from
number theory to finite field arithmetic may seem a bit daunting at
first, instructors and students will find that familiar concepts like
the Euclidean Algorithm and modular inverses carry over quite nicely
to the new setting. Furthermore, the generalization from numbers to
polynomials provides an excellent “bridge” for those students who will
being going on to an abstract algebra course, and an opportunity to
stretch (without breaking!) the minds of those students who might
not. This talk will focus on the “Simplified Advanced Encryption
Standard” (S-AES) which illustrates all of the features of AES at a
level of complexity which does not require the use of computers to do
examples.