1. Diffie-Hellman Key Exchange
Yuen Ng

2. Importance Allows a secret to be made between two people
Even if the whole world is listening in
This secret can be used as an encryption code word.
Particularly (and mainly) on the internet
Whitfield Diffie
Martin Hellman
(Ralph Merkle)

3. What’s the problem? For Alan and Betty…
Muahaha
For Alan and Betty…
Niamh is listening in on the communications
Alan and Betty want to form a secret password that only they know.
But, any password that Alan wants to tell Betty won’t be secret…
They can use this to encrypt sensitive information
Eg what city state they are going to attack, which credit card number she’s using

4. What’s the solution? Announced publicly in 1976
We can use gab as the new secret, the new code word
So we can now send long messages to each other, in private
British Signals Intelligence Agency by Malcolm J. Williamson but classified
We can easily make it bigger
Credit card details,
Using other encryption methods that rely on a shared secret; a different system that requires one key only

5. Diffie-Hellman Mechanism
(ga)b = (gb)a = gab
For example (23)6 = 86 = 262,144
(26)3 = 643 = 262,144
gab = 262,144
Everybody is taught this in secondary school
It’s easy to form powers of g
It’s not necessarily easy to find which numbers you used as the power of g (in the intermediate steps)

6. Forming a key with Diffie-Hellman
(ga)b = (gb)a = gab (mod p)
For example (237)10 = 3,404,825,44710 = 209,386,425… for 96 digits in total
(2310)7 = 41,426,511,213,6497 = 209,386,425… again
So gab = … * 10^95
But (mod 9719) we have
(237)10 = 3,404,825,44710 = = 1195
(2310)7 = 41,426,511,213,6497 = = 1195 again!
So gab = 1195
Everybody is taught this in secondary school
Since the numbers get large very very quickly, we can use mod arithmetic
Where we are just looking at remainders
By the way 9719 is a specially chosen prime
9719 is the largest four digit prime for which consecutive pairs and triples of digits are also prime (like 97, 19, 71, 971 and 719)
We still get the same answer in mod arithmetic, whichever way round the powers are applied
And the answer is much smaller and more manageable

7. Diffie-Hellman Key Exchange
(ga)b = (gb)a = gab (mod p)
To everybody: We choose g = 23 and p = 9719
A to B: here’s ga = 7053
B to A: here’s gb = 4800
PRIVATE
Alan:
(gb )a = 4800 a
= 1195
PRIVATE
Betty:
(g a)b = 7053 b
= 1195

8. Diffie-Hellman Key Exchange
(ga)b = (gb)a = gab (mod p)
To everybody: We choose g = 23 and p = 9719
A to B: here’s ga = 7053
B to A: here’s gb = 4800
PRIVATE
Alan:
(gb )a = 4800 a
= 1195
PRIVATE
Betty:
(g a)b = 7053 b
= 1195

9. Diffie-Hellman Key Exchange
(ga)b = (gb)a = gab (mod p)
To everybody: We choose g = 23 and p = 9719
A to B: here’s ga = 7053
B to A: here’s gb = 4800
PRIVATE
Alan:
(gb )a = 4800 a
= 1195
PRIVATE
Betty:
(g a)b = 7053 b
= 1195
I can’t work out from the information what the secret key is!
I’d love to know a and b.

10. Niamh is in difficulty It’s easy to form powers of g
g35 = (((((g2) 2) 2) 2) 2) * g2 * g
Suppose (like Niamh) you know the values of g, p, ga and gb. Can we work out gab ?
One way to solve: we could find a from ga and then find (gb)a = gab which is the secret
But, it’s not at all easy to find which numbers you used as the power of g (in the intermediate steps)
Repeated squaring technique
Five squarings, and three multiplications – faster than 34 operations (and memory storage)
Given that 7053 and 4800 are powers of 23 in mod 9719
Multiplying together won’t work – a + b

11. Security
Currently no known algorithms are efficient at solving the Discrete Logarithm Problem
Works well if you use large primes
Eg p = 2q + 1
(only 2 and q, both primes, divide the order of g)
Human neglect…
…and impostors
9719 is a very small prime. We use 768 and 1024 bit primes in practice – very large numbers to make it hard to guess
Could write out all powers of g, but fastest method is… and has time complexity…
How to impersonate (man-in-the-middle attack)
picture of impersonation
pictures
Log Jam
NSA
The Impostor’s own information