
Galois Field in Cryptography (2012) [pdf] - jpelecanos
https://sites.math.washington.edu/~morrow/336_12/papers/juan.pdf
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MrBingley
I'm taking a class in coding theory right now, and I've been surprised by how
naturally it can be described using abstract algebra. There are also many deep
relationships to important results in group theory. For example, the
automorphism group of the binary Golay code (which was used during the Voyager
missions to transmit pictures back to earth) is the Mathieu group M24, one of
the sporadic groups from the classification of finite simple groups!

[https://en.wikipedia.org/wiki/Binary_Golay_code](https://en.wikipedia.org/wiki/Binary_Golay_code)

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mozumder
Best part is Evariste Galois did all his field theory before he died at 20..
puts all other teenage mathematicians to shame.

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quiq
I'm in my second semester of abstract algebra, and my professor has mentioned
a few times that there should really be a movie made about Galois. "It's like
if Good Will Hunting was real, except better in every way, and instead of a
love story it's full blown Vive la France."

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userbinator
Good that this is more on the practical side, i.e. talking about bits and
bytes instead of just abstract numerical theory. It really helps when learning
this stuff --- a while ago I was reading about Reed-Solomon (which also uses
GF) and I could find plenty of theoretical material, but there was a
noticeable shortage of practical implementation-oriented detail.

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dnautics
Oh just be careful one of the early seminal "how to implement" papers has an
incorrect description of the vandermonde matrix which results in not being
able to guarantee matrix inversions. Be sure to check the errata for older
papers.

If you want a good "halfway between theory and practice" experience, I suggest
implementing gf256 in Julia. That is, create the datatype and define +, -, *,
/. (Along the way g^n and log_g might be helpful too). For Julia > 0.6.2 the
builtin lu factorization operator is general enough that once you've
definitely the basic 4 operations (and zero(gf256) and one(gf256)) you can
call the matrix solve operation \ on your datatype and immediately recover
your erasures without having to go through tedious coding of an elimination
routine.

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pyvpx
it's a shame Galois Fields are basically completely/thoroughly patented for
communications network use.

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amelius
I was under the impression that mathematics cannot be patented.

A case of proprietary protocols? (Like MPEG?)

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pyvpx
mathematics itself can't be patented, sure. but it seems their specific
application can be -- see:
[https://datatracker.ietf.org/meeting/92/materials/slides-92-...](https://datatracker.ietf.org/meeting/92/materials/slides-92-nwcrg-3/)

galois fields are very interesting for random linear network coding, so of
course there is a very thorough patent group (recently sold off, I believe)
around it.

given the seriously transformative effects it can have on communications
networks -- especially wireless -- I'm both saddened and unsurprised it's in
basically nothing as of yet. because it's patented to Hell & Back.

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techwizrd
I'm taking a class in abstract algebra as part of my graduate degree in math
and I'm constantly finding parallels in the way we code and represent programs
and data structures. Seeing the abstract theory in practical use like this is
always fascinating.

