1. Diffie-Hellman Diffie-Hellman is a public key distribution scheme First public-key type scheme, proposed in 1976.

2. Diffie-Hellman Public-key distribution scheme Cannot be used to exchange an arbitrary message Exchange only a key, whose value depends on the participants (and their private and public key information) The algorithm is based on exponentiation in a finite field, either over integers modulo a prime, or a polynomial field

3. Diffie-Hellman The algorithm –Alice and Bob agree on two large prime num, p and q. –Alice then chooses another large random number x and calculate A such that A=q ^ x mod p. and send to bob –Bob also chooses a another large num y and calculate B such that B=q ^ y mod p. and send to Alice –Both Alice and Bob can calculate the key as K 1 = B ^ x mod p K2=A ^ y mod p K1 = K2 –The key may then be used in a private-key cipher to secure communications between A and B

4. Diffie-Hellman Let p = 11 and q = 7 Alice chooses another num x = 3 then we have A = q^ x mod p =7 ^ 3 mod 11 = 2 Alice Sends the number A = 2 to Bob Bob chooses another num y = 6 then we have B =q^ y mod p = 7 ^ 6 mod 11 = 4 Bob sends the number B = 4 to Alice Now Alice generate Secret key, K1 =B ^ x mod p = 4 ^ 3 mod 11 =9 Then Bob generate Secret key, K2 =A ^ y mod p = 2 ^ 6 mod 11 = 9

5. Key Exchange: Diffie-Hellman Alice Bob A = g ^ x mod n A K1 = B ^ x mod nK2 = A ^ y mod n B B = g ^ y mod n

6. Mathematical Theory Behind Algorithm First Alice find key K1 = B ^ x mod n but what is B ? B = g ^ y mod n, therefore if we Substitute this value of B in K1 then K1=(g ^ y)^x mod n = g ^ yx mod n Then Bob find key K2 = A ^ y mod n but what is A ? A = g ^ x mod n, therefore if we substitute this value of A in K2 then K2 = (g ^ x)^y mod n = g ^xy mod n Now Basic Maths says that: K^ yx = K^ xy