25 August 2015

Random Thought

Reading through an introductory textbook on Cryptography tonight (this one, which is well-written at least as far as the first few chapters).  Reached the section on AES.  Before explaining how AES works, the authors run through the basics of Groups and Galois fields.  They run through the properties of Groups (closure under whatever function, associativity, having an identity element, invertibility).  I've thought a little in the past about closure, which is a very cool property.  But I hadn't thought much about associativity before.  It's funny, because they teach you associativity in your first algebra class.  It's a very basic thing.  But there it's grouped with all the other basic algebraic properties which are taught functionally as instruments for understanding the relations between terms and the order of operations implied by a given expression.  I was never taught to consider what it would mean for an algebraic system to be non-associative.  Obviously we encounter functions that are non-associative all the time (exponentiation, the cross product, etc.)...  Still, it's interesting.