Diffie-Helman example in g++

[hughes@ah.com (Eric Hughes)]

>Earlier, Douglas Barnes wrote:

>> // Demo of mathematics for Diffie-Hellman type key exchange
>[..]
>> // Does anyone have a clue what good values of 'a' are in this
>> // algorithm?
>> 
>> a = 127;

Notation: here 'a' is the base of the D-H exponentials.

>Feel free to correct me if I'm wrong, [...]

Certainly.  ;-)

>The only restriction placed on /a/ is that it be a primitive root of
>/p/. 

D-H works, i.e. a key is agreed upon, even if 'a' is not a primitive
root mod p, but the security may be adversely affected if it is not.

If 'a' is not a primitive root, then size of the search space which
the exponentials may take will be less than maximal.  In fact, the
order of the element 'a' gives the number of such possibilities.  (The
order is the smallest power of an element that is equal to the
identity.)

>To do this, you choose /a/ at random until you find the condition
>(/a/, /p/-1) == 1 is satisfied. 

Nope.  Being relatively prime to p-1 is not even involved.  Here is
the actual condition for primitivity:

   For every prime q which divides p-1, a^((p-1)/q) != 1 (mod p)

By Fermat's Little Theorem, x^(p-1) == 1 (mod p), for all 'x'.  Now
'a' is primitive if p-1 is the smallest such number.  Since the order
of an element much divide the order of the group, if no divisor d of
p-1 is such that x^d == 1 (mod p), then p-1 must be the smallest.

Burt Kaliski, of RSA Labs, told be he picked a D-H modulus p such that
p = 2q+1, where both p and q are prime.  It took a long time to find
such a pair.  The advantage is that almost half the elements of such
a field are primitive roots.

Eric