# irreducible Polynomial in a Cipher

If we change the irreducible polynomial of the AES to someother irreducible or primitive polynomial, what effect will it have in security?

The polynomial in the AES is $x^8+x^4+x^3+x+1$ that is irreducible but not primitive. In fact, you can use an irreducible and primitive polynomial but you have to save 256 cases in computation. Another main reason for using $x^8+x^4+x^3+x+1$ in AES is mix-column state. In mix-column state we need an MDS matrix. The best optimal MDS matrix is circular MDS matrix that is constructed with the third element of finite field ($1 \,,\, \alpha \,,\, \alpha +1$). If you want to use an irreducible and primitive polynomial, you have to check that is there an optimal MDS matrix like MDS matrix of AES. I think, AES is designed such that with minimum RAM can be applied and has a proved security. In other words, in min-max problem of cryptography (security and speed) AES is an optima algorithm.