Can the AES encryption and decryption algorithms be described as a polynomial for a specific key? AES is a block cipher that consists of non-linear and linear parts to provide the Shannon requirements of confusion  and diffusion respectively. The linear parts can for sure be described as a polynomial and more specifically the SiftRows,MixColumnds,AddRoundKey parts but what about the non-linear one?
Yes, they can be described as a multivariate polynomial over $GF(2)$ (or over $GF(2^8)$).  See algebraic cryptanalysis.  This expression does not seem to help cryptanalyze AES, so far as we know, but it can be done.

For an example of how to write AES in this way, see the following paper:


A simple algebraic representation of Rijndael.  Niels Ferguson, Richard Schroeppel, Doug Whiting.  SAC 2001.


You can also read about the XSL attack, a controversial attack that was claimed by some researchers to work against the AES and which is based upon a representation of AES as multivariate polynomials.