Some stuff about Diffie-Hellman (and more :-)

[rcain@netcom.com (Robert Cain)]

In the Diffie-Hellman exchange there is a well-known-prime, w, and a
well-knwon-modulus, m.  For those interested that don't know I think
it then proceeds as follows (don't have notes in front of me so please
someone correct me if I'm misremembering it) where ** is the power or
exponentiation operator and % is the modulus operator:

	1) Bob generates a one time random prime, b, then computes
		B = (w ** b) % m
	   and sends B to Carol.

	2) Carol generates a one time random prime, c, then computes
		C = (w ** c) % m
	   and sends C to Bob.

	3) Bob generates a session key:
		K = (B ** c) % m

	4) Carol generates a session key:
		K = (C ** b) % m

Carol and Bob have the same K because:
	K == (C ** b) % m == (B ** c) % m == (w ** (b * c)) % m

From just the knowledge of B and C a snoop cannot determine
b from B, within computational reason (the root modulus being as
difficult as factoring), nor c from C, and because K cannot be 
determined from B and C without knowing b or c, she is screwed.

Now, the tutorial over :-), the question is; is there a "standard"
well-known-prime, w, and a "standard" well-known-modulus, m, and if
not, let's define one.  I suppose that PGP uses a well known pair but
they are big and not easy to hand around without going through media (I
think.)  When defined algorithmically they might be easier to actually
incorporate in a program or a product than great big numbers.  If this
has not been done, I propose a simply stated algorithm for finding a
"standard" w and m that will allow interoperation among all future
implementations of D-H as follows:

	Let "standard" w be the first prime found probing from the
	starting point w' = n!, with a well-known n that should be
	small. I am not sure what n should be to generate a large
	enough w'.  Let's just say the smallest n that generates a 1000
	digit number.  There is a well known primality testing
	algorithm by Lenstra that is pretty agreed upon by the number
	theory crowd (I have it coded by Lenstra and more on that
	later.) So, let w be the first number larger than w' that
	passes Lenstra's primality test.  Any program or device
	employing D-H will have this algorithm in it somewhere for
	generating each session specific b and c so all we need to
	agree on is 1000 (or whatever is decided to be a large enough
	prime for all practical purposes.)

I leave a "standard" for m up for discussion because I don't have
the material in front of me that tells the criterion for selecting
strong m's and there are some considerations.  I would like it to
be algoritmically defined though using standard long modulus, long
integer arthmetic and some small, easy to remember number.

Whatcha think?

Oh, for those of you that actually code this stuff like me, I have
Lenstra's long integer function package in C that I "ported" from K&R
to ANSI and edited and reorganized the documentation in the process.  I
interacted with him in that process and it is a stable and reliable
package.  This was a year ago so he has most likely added to it by now
but this snapshot I have is very complete and has way more than is
needed to do nearly anything in crypto.  And it is by Lenstra himself!
A cool guy BTW.  The problem:  I did have to make some changes to
macros and sundry things to ANSIfy it and may have introduced errors.
It runs his demonstration programs that are part of the package and
gives the correct results and these programs exercise a good part of
it, especially the areas I had to mess with.  BUT: I have not had the
time to sit down and look hard at a true verification suite and he
doesn't have one either. So, caveat emptor, I offer this package (and
the original from which it was derived) to *one* person that can put it
in a relevant ftp site.  Is that you, Sameer?

BTW, D-H is useless across a medium in which there can be an active
snoop or spoof as I guess we call him.  Whit, Marty and Ron agree as of
a discussion a year ago.  The spoof just has a pair of boxes and
separately negotiates a session with Bob and one with Carol so that
clear text passes between his pair.  There is no way in theory to
detect the presence of our friendly spoof.  :-)

I've found a solution to this that is more than sufficiently secure in
practice and even theoretically secure in most practical situations.
I'm not sure what to do with it.  I would like to retire on it though
(and get a couple "voluntary income tax" liens off my back :-) and
perhaps even endow some kind of institute.  Actually I worry more about
being retired because of it if you get my paranoid drift.  I guess that
is why I'm lettin' y'all know about it here first.  I am also curious
about how you folks here feel about someone wanting to personally
benefit financially from an algorithm/protocol invention/discovery like
this but I don't want nor will get into any flame war.  :-(


Peace,

Bob

-- 
Bob Cain    rcain@netcom.com   408-354-8021

 "Morality is largely a rationalization of the  point you happen to occupy
  in the power pattern at a given time.  If you're a *Have-Not* you're out
  to *get*, and   your morality is an appeal to a law higher than man-made
  laws--the noblest  ideals of  justice and  equality.  When you  become a 
  *Have* then you are out to *keep* and your morality is one of law, order
  and the rights of property over other rights."

                                                       Saul D. Alinsky
                                                          1909-1972

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