Re: pmult

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Fri, Nov 12, 2021 at 04:19:03PM +0100, Philippe de Rochambeau wrote:
> Very enlightening, Bill. Many thanks.
> 
> If GF(2^8) aren’t polynomials, what are they?

They can be identified with class of polynomials modulo some irreducible
polynomial of degree 8 over GF(2). In the same way that the elements of
GF(2) are not the integers 0 and 1, but their classes modulo 2.

? X=ffgen((x^8 + x^4 + x^3 + x + 1)*Mod(1,2),'X)
? X^10
%2 = X^6+X^5+X^3+X^2

Clearly x^10 and x^6+x^5+x^3+x^2 are different polynomials,
but they have the same class modulo x^8 + x^4 + x^3 + x + 1.

Indeed:
? x^10 % ((x^8 + x^4 + x^3 + x + 1) * Mod(1,2))
%3 = Mod(1,2)*x^6+Mod(1,2)*x^5+Mod(1,2)*x^3+Mod(1,2)*x^2

Cheers,
Bill.