Diffie Hellman - logs in Galois fields

[paul@fatmans.demon.co.uk]

Hi all,

A question for the matematicians out there:

I am looking at the Diffie Hellman public key exchange protocol and 
am trying to find out why it is computationally hard to take logs in 
a finite (Galois) field.

My maths tutor has told me a bit about the construction of Galois 
fields (If I`m correct the construction is Z mod N, N some integer, 
then a transformation F(x) on the residue classes already in the 
field) I know also the definition is that there are P**k elements, p 
a prime.

My questions are as follows:

1. How can a field be finite, as by definition it has to be closed 
under addition, subtraction, multiplication and division???? (sorry 
if this one is a bit of a no brainer, maybe the definition is 
different but I can`t seem to see how)

2. Why is taking logs in a finite field computationally hard? - Me 
and Alec (My maths tutor at college) guessed that it is because 
exponentiation and logs are each others inverse functions, and 
somehow this becomes a one way function in a finite field.

3. Are the Galois fields used in Diffie Hellman specially constructed 
in any way or are they just normal GF????

Thanks for any help anyone can give me....


  Datacomms Technologies web authoring and data security
     Paul Bradley, Paul@fatmans.demon.co.uk
       Http://www.fatmans.demon.co.uk/crypt/
         "Don`t forget to mount a scratch monkey"

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