Cool article overall, especially since it was an actual CS related application of the abstract algebra class I took this semester. Also, is there any reason in particular why that particular polynomial was chosen for AES?
from the Rijndael submission to the AES competition, page 25 (http://csrc.nist.gov/archive/aes/rijndael/Rijndael-ammended....)
LiNi86 refers to is R. Lidl and H. Niederreiter, Introduction to finite fields and their applications, Cambridge University Press, 1986.
So there is no apparent reason they choose this polynomial, i.e. other would also have worked.
"That first one that worked (100011011) is the one used in AES"
i.e., the polynomial that was chosen for AES (x^8 + x^4 + x^3 + x + 1) is the minimal irreducible polynomial of degree 8 (minimal in the sense of the natural ordering of polynomials). I guess that could be a good reason (but I'm not sure it's _the_ reason).