Galois Field in Cryptography (2012) [pdf]

I&#x27;m taking a class in coding theory right now, and I&#x27;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!<p><a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Binary_Golay_code" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Binary_Golay_code</a>

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

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.

it&#x27;s a shame Galois Fields are basically completely&#x2F;thoroughly patented for communications network use.

I&#x27;m taking a class in abstract algebra as part of my graduate degree in math and I&#x27;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.