---------------------------------------- firstname.lastname@example.org (Steve Pope) writes: > Path: news.mathworks.com!newsfeed-00.mathworks.com!news.kjsl.com!wasp.rahul.net!220.127.116.11.MISMATCH!rahul.net!not-for-mail > Newsgroups: sci.math,comp.dsp > Subject: Re: Polynomial used to create Galois field for AES? > Date: Thu, 29 Apr 2010 13:36:22 +0000 (UTC) > Organization: a2i network > Originator: email@example.com (Stephen P. Pope) > > Jaco Versfeld <firstname.lastname@example.org> 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).
This is by design in AES, which uses two GF(2^8) polynomials and other operations to generate the SBOX for the algorithm. The reason an irreducible but not primitive polynomial is used is that we are trying to make a non-linear permutation function that has diffusion, spreading input bits to output bits in an non-linear way. The AES sbox was analyzed to death during the competition to pick the AES algorithm.
Here is some MATLAB code I have used in the past to generate the AES SBOX--the gf arithmetic requires Comm Toolbox in MATLAB:
temp = gf(0:255,8,283).^254; % inverses in GF(2^8) with polynomial 283 temp = gf(temp.x,8,257).*31; % now embedded that field into poly=257 field and mult by 31 sbox = bitxor(temp.x,99); % now bitwise xor with 99 sbox = double(sbox);