# If we use another irreducible polynomial for AES how can we show it is still resistant to DCA and LCA?

AES has been rigorously tested against all known attacks. However, I believe there are 30 suitable irreducible polynomials that could be used for AES. If we were to change the current irreducible poly to another suitable one:

1. how can we be sure it will still be protected against DCA and LCA attacks?
2. can we show it is still protected against DCA, LCA and other attacks without having to test large amounts of data? If so, how?
3. how, if at all, would this change other parts of the cipher (e.g. MixCols)

I have tried to find clear answers on this but they seem hard to come by. Many books have merely asserted that changing the irreducible poly (to one of the other 29) would not adversely affect security, but I cannot find a clear reasoning.

A simple answer with clear examples would be much appreciated. I am still very much a beginner in cryptography.

• As far as I can tell, changing the polynomial does not change the nonlinearity or differential uniformity of the s-box (I have tested all polynomials) – Richie Frame Oct 12 '17 at 23:58
• When you say you have 'tested all polynomials', do you mean you have run computer programs? Is there a simple mathematical way to show they are all suitable? I understand they are all isomorphic but how can we use this to show how they are all equally (or more or less equally) resistant to DCA and LCA, for example? – Red Book 1 Oct 13 '17 at 14:15
• yes i computed all valid s-boxes from all polynomials, and they all had the same NL and DU values when valid s-boxes were generated, what did change was classical properties such as avalanche and bit independence – Richie Frame Oct 15 '17 at 23:18
• Were these changes merely different from the original but just as secure? Or where they less favourable in terms of security? I wonder if there is a simple way to explain (without test results) why any of the 30 polynomials would produce equally secure ciphers as the original. Perhaps in terms of the polynomials in question being isomorphic to one another. – Red Book 1 Oct 16 '17 at 7:00