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So, was that reply intended to be helpful and to provide the sought-for explanation, or only intended to make you feel good about yourself?

The former, Gareth.

If you've found it helpful then I will also be pleased to have helped you.

It could be both, maybe making people feel good about themselves IS the only practical application :)

You seem to be confusing me with someone else and are replying with a response that seems to continue to be more of an irrelevant ego trip than to be a helpful response.

Insofar as you do confuse me with someone else, are you the same Brian Reay who lives at Falcon Lodge, Spekes Rd, Hempstead, Gillingham, Kent, ME7 3RT and who has the CB callsign of G8OSN?

Well, I've a couple of school maths texts here and the chapters on group theory are the only ones with no seeming practical application.

How would you do a calculation with the theory that would give you a useful result in those areas?

(I don't expect a mathematical treatise, just perhaps an example of how it is applied)

Or to put it another way, if you were to totally ignore the group theory in those applications, how would you be disadvanteged, if at all?

In article , Amateur Machinist writes

Well, it was several decades ago and I found it hard to stay awake (was never my favourite area of chemistry) but as far as I can recall it had to do with the fact that the properties, energy levels, spectroscopic patterns etc. of atoms depended on the symmetry of their molecular environment. Don't ask me how the calculations are done, I did a few at the time but the years have mercifully blanked it out.

David

....sigh...here we go again. When do the schools go back...? If you guys have a peeve..take it to e-mail or the playground please. --

Chris Edwards (in deepest Dorset) "....there *must* be an easier way!"

Well, Galois theory is useful to understand symmetry of roots to polynomials- the most basic example being that quadratic equations have either one (repeated or double) root, two real roots, or a pair of complex roots. You could survive life without knowing such things but were you, for example, an electronics engineer, such things do tend to be useful.

HTH

Brian

No "peeve" here Chris. The poster asked a question and got an answer.

Regards

Brian

So does Galois go on the tumbler gear or does the polynomial ? And if the polynomial goes on the tumbler gear do you need a root, two real roots, or a pair of complex roots between the Galois ?

If it's the first option where does the kipper fit ?

Confused minds want to know.

Is that near the sewage-works?

Given Derek the night off, Beanie?

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

Oh, and as to me personally using groups and group theory - the method in this 2004 paper is I think is unique in cryptography in using a nested set of four groups, each a proper subgroup of all the higher groups.

-- Peter Fairbrother

An interesting and informative post Peter, which I enjoyed reading. Thank you.

That reminds me of a something I was told re Object Oriented Programming.

Used, if I recall, to establish a secure code between to essential strangers. They pass (in the clear) some basic numbers (inc a Prime) and using index laws and modulo maths can establish a secure code. Not looked at it in some time but I'm pretty sure that is the basics.

Thank you again.

Brian

But that is a lousy approximation for pi and the end of the link :P

Yep.

I don't often talk about crypto on non-crypto fora, 'cos I think it's fantastically interesting but most people just go uh?, but - Diffie-Hellman really is astounding. That two people can establish a secret, openly, and an observer can't deduce the secret is just amazing.

Some other amazing things you can do with crypto: You can query a database and get an exact number of bits from the database - but the database operator can't tell which bits of the database you got.

So you can look up something in a database and no-one can tell what you looked up. It's deniable too, delete the numbers behind the query once it's answered, and even you can't tell what it was.

You can also get a database to count the number of times a word or phrase occurs in the database, without the database knowing what the word is!

Digital signatures and certificates you probably know about, but they were only discovered in the 1970's, and they are pretty amazing too.

Then there's steganographic file systems, which hide the number and sizes of the files they may contain, and, relying on distributed trust rather than mathematics, there are mixnets which can defeat traffic analysis, hidden servers (where no-one can tell where the server is, but you can still get a page from it) and .. I'll stop here.

The problem today is that most of the actual implementations are cr*p.

Robert Morris's (ex-NSA, author of the Unix "crypt" library) rule one of cryptanalysis, "First look for plaintext", holds now more than ever - it's usually a lot easier to find plaintext than to break the crypto.

People just don't encrypt for whatever reason, even when they should. Operating systems generate copious temporary copies of many files, which are seldom if ever securely deleted.

Red/black separation (separating encrypted and plaintext signals) is hard, and seldom done correctly - people pWn machines all the time.

Keys are too short and subject to various attacks, including brute force (trying all possible keys), rubber hose ("Give me the key and I'll stop beating you") and the nice truncheon (aka RIPA) ("Give me your keys or I'll send you to jail for x years" - which doesn't need a Warrant or a Court Order, a Policeman issues the demand) attacks.

Some effort is made to prevent man-in-the-middle and other protocol-based attacks, but almost all present systems can be beaten using this type of attack - it's just that people don't bother.

Good modern ciphers are probably unbreakable by man today, and it's not too hard to implement one which will be almost certainly unbreakable ever, except maybe by God, though again few people bother.

And why should they? It's so much easier to get the data by breaking the system than breaking the crypto.

Though we shouldn't forget Robert Morris's other maxim: "Never underestimate the attention, risk, money and time that an opponent will put into reading traffic".

I'll stop here, really this time.

-- Peter Fairbrother