On Sun, 28 Dec 2008 12:41:43 -0000, "Amateur Machinist"

....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!"

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.
http://www.springerlink.com/content/q07439n27u1egx0w/?p dee7597a5b4ab2b9c5797c35b22044&pi=4
It's what I do.
-- Peter Fairbrother

<snipped for brevity>
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.
>> In the Diffie-Hellman group for instance, the set is the integers less

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
www.g8osn.org.uk

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

On Wed, 24 Dec 2008 07:18:52 -0000, "Amateur machinist"

Quite so, allowing for the fact that neither miles nor gallons are in
m^3, but the principle is correct. The 'error' as such would be a
simple ratio (dim-less number) of miles/m per gallons/m^3. It can be
useful in checking that you've got relationships in equartions
correct. As noted earlier, both sides of an equals sign must have the
same dimensions.
When working with any of the heap of dimensionless numbers (eg
Reynolds, Mach, Froude, Nusselt etc etc) or some of the more obscure
units (viscoscity gives some of the most strange, being
pressure-seconds) totting up the string of dimensions and checking
that they do cancel to the desired result is a quick easy means to see
if you've made a basic clanger. More subtle clangers are still
entirely possible;-)
Richard

There are some unobscure derived quantitiess which when expressed in
SI base units can look incromprehensible. Try these two.
1. m^2 kg s^-3 A^-2
2. m^-2 kg^-1 s^4 A^2

That would be dimensionally correct. Of course, you could equally well
quote sfc in litres per kilometre (hence having dimensions L^2) or
kilograms per kilometre (ML^-1) or even joules per kilometre (MLT^-2).
Such conversion factors necessarily take their dimensions from the units
chosen for their definition; any attempt to relate them to other
meaningful parameters will likely drive you nuts. Richard's recent post
is entirely in point here.
David

Yes, it resolves to 1/length^2, and so can be expressed in reciprocal
acres or reciprocal ares. Alternatively, following the convention
established for conductivity, in ercas or eras.
Regards,
David P.

I hope you are aware of the old, but emerging contender to replace the
ISO (MKS) system towit the FFF system ie Furlong, Firkin, Fortnight.
With a distance, a mass and a time, ALL other units can be derived.
Gives rise to some delightful but concise units eg the earlier mpg
becomes furlongs/firkin, - I doubt I could manage the reciprocal
firkin/furlong, but I'd be willing to try as long as it was Speckled
Hen or HSB <G>
Richard

Polytechforum.com is a website by engineers for engineers. It is not affiliated with any of manufacturers or vendors discussed here.
All logos and trade names are the property of their respective owners.