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Topic: Polynomial used to create Galois field for AES?
Replies: 8   Last Post: Aug 11, 2010 3:46 PM

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 Steve Pope Posts: 30 Registered: 9/5/05
Re: Polynomial used to create Galois field for AES?
Posted: Apr 29, 2010 9:36 AM

Jaco Versfeld <jaco.versfeld@gmail.com> wrote:

>The following polynomial is used to create a "Galois field" GF(2^8)
>which is specified in the Advanced Encryption Standard (AES): p(x) =
>x^8 + x^4 + x^3 + x + 1.
>
>However, I checked the polynomial (quickly using Matlab) whether it is
>primitive. It turns out not to be primitive, but still irreducible (I
>haven't yet confirm this for myself, though).

Yes, I just checked it, and it's not primitive, and it does not
generate GF 256.

>According to my knowledge, you need a primitive polynomial in order to
>construct a "proper" Galois field (or extended Galois field).

>Why can we use an irreducible polynomial for AES, would it not cause
>problems (not every element will have an inverse, similar as when we
>construct a ring mod x, where x is a composite)?

Depends how they used it. I do not know where it is in the AES
algorithm. If it's part of just generating a keystream or hash
function, nobody cares if it isn't primitive. But you could not use
it for coding however.

In any case you're right this poly does not generate the field.

Steve

Date Subject Author
4/29/10 jaco.versfeld@gmail.com
4/29/10 Timothy Murphy
4/29/10 dvsarwate@gmail.com
4/29/10 dvsarwate@gmail.com
4/29/10 jaco.versfeld@gmail.com
4/29/10 Steve Pope
4/29/10 Brian Ogilvie
4/30/10 I.M. Soloveichik
8/11/10 Larry D'Anna