1. Information Security and Management 10. Other Public-key Cryptosystems Chih-Hung Wang Fall 2011 1

2. Diffie and Hellman 1976 A number of commercial products employ this key exchange technique This algorithm enables two users to exchange key securely 2 Diffie-Hellman Key Exchange

3. 3 Algorithm of Diffie-Hellman (1/2)

4. 4 Algorithm of Diffie-Hellman (2/2)

5. 5 Example of D-H Key Exchange q=97 =5X A = 36 X B =58 Y A =5 36 =50 mod 97 Y B =5 58 =44 mod 97 K=(Y B )X A mod 97 = 44 36 = 75 nod 97 K=(Y A )X B mod 97 = 50 58 = 75 nod 97

6. 6 Diffie-Hellman

7. RSA based hybrid encryption system 7 Supplementary (1) A B Randomly selects a DES Key (or other symmetric encryption system) K DES E K DES (M), E K puB (K DES )

8. Diffie-Hellman based hybrid encryption system 8 Supplementary (2) A B YAYA YBYB K=(Y B ) xA =(Y A ) xB Mod q SK=h(K) 128 – 256 bits E SK (M)

9. In 1984, Elgamal announced a public-key scheme based on discrete logarithms. Closely related to the Diffie-Hellman technique. ElGamal Cryptographic System 9

10. q : prime number α: α<q and α a primitive root of q Global Public Elements 10

11. Select private X A : X A <q-1 Calculate Y A : Y A = α X A mod q Public key: PU={q, α, Y A } Private key: X A Key Generation by Alice 11

12. Plaintext: M <q Select random integer k: k<q Calculate K: K=(Y A ) k mod q Calculate C 1 : C 1 = α k mod q Calculate C 2 : C 2 =KM mod q Ciphertext: (C 1, C 2 ) Encryption by Bob with Alice’s Public Key 12

13. Ciphertext: (C 1, C 2 ) Calculate K: K=(C 1 ) X A mod q Plaintext: M=(C 2 K -1 ) mod q Decryption by Alice with Alice’s Private Key 13