
Visualizing Galois Fields  - wglb
http://nklein.com/2012/05/visualizing-galois-fields/
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dude_abides
Wolfram demo visualization of finite fields:
<http://demonstrations.wolfram.com/FiniteFieldTables/>

(requires the Wolfram CDF player)

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xyzzyz
_Just before the above-mentioned article hit reddit, I got to wondering if the
structure of the Galois field was affected at all by the choice of polynomial
you used as the modulus._

Not really -- regardless of a choice of an irreducible polynomial for a
modulus, all the fields you'll obtain are isomorphic, meaning that for two
fields K, L with 2^n elements there a one-to-one correspondence f: K -> L,
such that f(a+b) = f(a) + f(b) and f(ab) = f(a)f(b). This means that while K
and L may be different as sets, they have essentially the same structure.

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timb
that first image looks like the old munching square effect
<http://en.wikipedia.org/wiki/Munching_square>

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jasondavies
It also reminds me of <http://www.jasondavies.com/hamming-quilt/> where the
Hamming distance is used.

~~~
hendzen
Yes, the munching square picture is the same because the hamming distance of
two binary numbers is their XOR.

As to why it looks similar to the visualization in the article, your hamming
distance picture is essentially the multiplication table for a galois ring
GF(2^n), where n is width of the bits of the binary numbers you were using.

Note that I say galois ring, not field; assuming that an n (for n even) bit
unsigned binary number overflows as such: 0x00..01 + 0xFF..FF = 0x00..00, then
your chart is the multiplication table of the ring formed by taking Z_2 modulo
the ideal generated by the polynomial <x^n-1 + x^n-2 + ... + x^3 + x^2 + x +
1>, which is a ring since this polynomial is reducible (it has a root of 1).

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wging
A quibble of terminology: fields are rings too, so what you want to say is
that it's _not a field_ (or that it's 'only a ring') because the polynomial is
reducible, rather than that it's a _ring_ because the polynomial is reducible.
It's a ring because it satisfies all the necessary properties--whether the
polynomial is reducible or not has no bearing at all on whether the object
under discussion is a ring.

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hendzen
Cool article overall, especially since it was an actual CS related application
of the abstract algebra class I took this semester. Also, is there any reason
in particular why that particular polynomial was chosen for AES?

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dlsym
I think it was chosen because the addition of two elements in GF(2^8) with
their binary representation is a XOR. Which is a really efficient operation.

~~~
hendzen
Yes, but this is true regardless of the irreducible polynomial used to form
the field. My question is why that specific polynomial was chosen; was it an
arbitrary choice or was there some reason?

~~~
tomerv
As the article mentions:

"That first one that worked (100011011) is the one used in AES"

i.e., the polynomial that was chosen for AES (x^8 + x^4 + x^3 + x + 1) is the
minimal irreducible polynomial of degree 8 (minimal in the sense of the
natural ordering of polynomials). I guess that could be a good reason (but I'm
not sure it's _the_ reason).

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mrmagooey
Oddly, the first set of images looks like the visual 'aura' effect that I get
when I get a migraine. I immediately had to stop read the article
unfortunately, the association was quite strong.

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conformal
i think the title of the article is misleading, it should have been something
like "quick Galois Field overview". 3 or 4 images of GFs is not "visualizing"
them.

take a commutative ring, mod out by a prime ideal and you've got a field.
congratulations.

~~~
t-ob
Modulo a prime ideal you're only guaranteed an integral domain. The residue
ring is only a field when the ideal you're taking as a modulus is maximal. All
maximal ideals are prime, but the converse is not true - for example in the
ring Z[x] of polynomials with integral coefficients the ideal (p) generated by
a rational prime p is prime but not maximal.

There are of course many rings in which all prime ideals are maximal. The
integers, for example, and more generally the integer ring of any number field
(which are not always principal ideal domains, but are examples of Dedekind
domains).

