On Wed, Mar 13, 2019 at 12:12 PM Friedrich Wiemer
<friedrichwie...@gmail.com> wrote:
>
> I discussed this again with my colleagues and maybe its just not so sure what
> the correct way to do is?
>
> Is it actually clear what the "correct" ordering of finite field elements is?
> The canonical ordering is 0, 1, a^1, a^2, ..., - but then this representation
> and ordering depends on the representation of the actual instance, so which
> polynomial is picked.
I'm not even sure whether a primitive element is always available in a
Sage finite field. A field might be constructed from any irreducible
polynomial, and IMHO
finding a primitive element is not very quick for large fields.
> If instead the elements are ordered after the representing polynomial is
> chosen, we get 0, 1, a, a+1, ... - but for example list(GF(2^3)) is
> differently ordered, depending on the chosen implementation. pari and ntl
> results in 0, 1, a, a+1, ... while givaro gives 0, a, a+1, ..., 1.
>
> Regarding this, it might be ok to work with the output of `sorted(GF(...))`,
> as its done currently. Nevertheless, there remains the problem with different
> polynomials for representing the finite field and thus the resulting S-box
> might be different. Here is an example of what I mean:
>
> sage: F1 = GF(2^3, name='a', modulus=PolynomialRing(GF(2), 'a')('a^3 + a +
> 1'))
> ....: F2 = GF(2^3, name='a', modulus=PolynomialRing(GF(2), 'a')('a^3 + a^2 +
> 1'))
> ....: R1 = PolynomialRing(F1, 'x')
> ....: R2 = PolynomialRing(F2, 'x')
> ....: inv1 = R1.gen()**(2**3-2)
> ....: inv2 = R2.gen()**(2**3-2)
> ....: S1 = SBox([inv1(v) for v in sorted(F1)])
> ....: S2 = SBox([inv2(v) for v in sorted(F2)])
> ....: S1, S2
> (0, 1, 5, 6, 7, 2, 3, 4),
> (0, 1, 6, 4, 3, 7, 2, 5)
>
> OK, so not so sure if this all makes sense in the context of the above
> question, but this behaviour should at least be mentioned in the docs, I
> think.
> Regarding the above discussed point, I still think that the current behaviour
> is 'wrong' in the way that one would expect a different result.
>
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