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. 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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