1. Diffie-Hellman key exchange/agreement algorithm

2. Introduction
It is provide the solution to the problem of key agreement or key exchange.
Two party agree on symmetric key. This key can be used for encryption/decryption.
This algorithm only for key agreement not for encryption and decryption.

3. Description of the algorithm
Suppose Alice and Bob want to agree upon a key to be used for encryption / decryption message that would be exchanged between them. Then diffie-hellman key exchange works as follows:
1. Alice & bob agree on two large prime numbers: n and g
(note: these two integer need not be kept secret, insecure channel)
2. Alice choose another large random number x and calculate A such that,
3. Alice send the number A to Bob.
4. Bob independently choose another large random integer y and calculate B such that,
5. Bob send the number B to Alice.
6. Alice, now compute the secret key K1 as follows:
A = gx mod n
B = gy mod n
K1 = Bx mod n

4. Description of the algorithm
7. Bob, now compute the secret key K2 as follows:
8. Surprise, K1=K2=K (which is symmetric key)
K2 = Ay mod n

5. Example of the Diffie -Hellman algorithm
Large Prime Number
Alice
n=11,g=7
Bob
Large Random
Number
x=3
y=6
Large Random
Number
A = 73 mod 11
= 2
B = 76 mod 11
= 4
B = 4
A = 2
K1 = 43 mod 11
= 9
K2 = 26 mod 11
= 9
K1 = K2 = K
Note: A,B,K1,K2 are Private to others

6. Mathematics Theory behind the algorithm
1) Alice perform K Now what is B?
If we put B value into K1 then,
2) Bob perform K Now what is A?
If we put A value into K2 then,
Now Basic Mathematics says that,
K1 = Bx mod n
B = gy mod n
K1 = (gy mod n)x mod n
K1 = gyx mod n
K2 = Ay mod n
A = gx mod n
K2 = (gx mod n)y mod n
K2 = gxy mod n
Kyx = Kxy so that K1=K2=K (Proof)

7. Problem with the Algorithm
Can diffie-hellman solve our problem associated with key exchange? Unfortunately, not quite!
This algorithm fall into “man in the middle attack”
This work as follows:
Step-1:
Alice want to communicate to Bob so as usual they use diffie-hellman key exchange so Let n=11 & g=7. they find K1 and K2 .
Step-2:
Alice does not realize that the attacker TOM is listening the conversation between her and Bob.
TOM simply pick up the value of n and g. (n=11,g=7)
Alice TOM Bob
Step-3:
Now let us assume that Alice, TOM & Bob select random number x & y as bellow:
Alice TOM Bob
n=11,g=7
n=11,g=7
n=11,g=7
x=3
x=8,y=6
y=9

8. Problem with the Algorithm
Step-4:
Now all these person calculate the value of A and B.
A=gx mod n A=gx mod n B=gy mod n
=73 mod 11 = =78 mod 11 = =79 mod 11 = 8
B=gy mod n
=76 mod 11 = 4
Step-5:
Alice TOM Bob
A=2
TOM intercept
The value of A
And send Bob his
Own value of A
A=9
TOM intercept
The value of B
And send Alice his
Own value of B
B=4
B=8

9. Problem with the Algorithm
Step-6:
So at last value of A,B to users are:
* indicate value of A and B changed.
Step-7:
Based on these values, all the three person now calculate their Keys.
A=2,B=4*
A=2,B=8
A=9*,B=8
Alice
TOM
Bob
K1=Bx mod n
=43 mod 11 =9
K1=Bx mod n
=88 mod 11 =5
K2=Ay mod n
=26 mod 11 =9
K2=Ay mod n
=99 mod 11 =5
Solution: ?
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