Re: imperial screwcutting on metric lathe

Amateur machinist wrote:
"Peter Fairbrother" <zenadsl6186@xxxxxxxxx> wrote in message news:495696fd$0$1331$fa0fcedb@xxxxxxxxxxxxxxxxx

(an old fart who's forgotten most of the calculus once literally beaten into him, but who does advanced math, number theory, group theory and so on (almost) every day)

Could you expund on what is your own practical application
of Group Theory, and how you actually do meaningful calculations
with it?

I'm a cryptologist, and many ciphers use groups, particularly but not exclusively public key ciphers.

For instance RSA uses the multiplicative group of invertible integers modulo PQ, where P and Q are primes, and Diffie-Hellman key agreement uses the group of integers modulo a prime.

Groups are less common in symmetric ciphers, in fact there are good reasons to ensure they are not groups under composition, but groups are not unknown - eg Pohlig-Hellman is a group, so (in a sense) is the one-time-pad and stream cipher, and there is some effort being made to create a secure cipher which is a group, though not much progress has been made as yet.

A detailed discussion of group theory is out of place here, but - a group is a set of objects, often numbers, combined with an associated binary operation which can be performed on any two members of the set, which also follows four rules:

there is an inverse for every element of the set,
there is an identity element,
the operation is associative and
the group is closed.


In the Diffie-Hellman group for instance, the set is the integers less than a prime, and the binary operation is multiplying two of them together to get a result modulo the prime.

Groups have some interesting properties, which is why we study and use them. They lead on to the study of rings and fields etc, and provide a sideways entrance to the study of arithmetics, more usually approached from the axiomatic perspective.

But that's mostly pure math, rather than more-useable stuff - though it's surprising how often "pure" math turns out to be useful and used.



-- Peter Fairbrother
.

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