Key Management in Asymmetric Encryption for Enhanced Network Security

1. CSCE 790:Computer Network Security Chin-Tser Huang huangct@cse.sc.edu University of South Carolina

2. Key Management • Asymmetric encryption helps address key distribution problems • Two aspects • distribution of public keys • use of public-key encryption to distribute secret keys

3. Distribution of Public Keys • Four alternatives of public key distribution • Public announcement • Publicly available directory • Public-key authority • Public-key certificates

4. Public Announcement • Users distribute public keys to recipients or broadcast to community at large • eg. append PGP keys to email messages or post to news groups or email list • Major weakness is forgery • anyone can create a key claiming to be someone else and broadcast it • can masquerade as claimed user before forgery is discovered

5. Publicly Available Directory • Achieve greater security by registering keys with a public directory • Directory must be trusted with properties: • contains {name,public-key} entries • participants register securely with directory • participants can replace key at any time • directory is periodically published • directory can be accessed electronically • Still vulnerable to tampering or forgery

6. Public-Key Authority • Improve security by tightening control over distribution of keys from directory • Has properties of directory • Require users to know public key for the directory • Users can interact with directory to obtain any desired public key securely • require real-time access to directory when keys are needed

7. Public-Key Authority

8. Public-Key Certificates • Certificates allow key exchange without real-time access to public-key authority • A certificate binds identity to public key • usually with other info such as period of validity, authorized rights, etc • With all contents signed by a trusted Public-Key or Certificate Authority (CA) • Can be verified by anyone who knows the CA’s public key

9. Public-Key Certificates

10. Distribute Secret KeysUsing Asymmetric Encryption • Can use previous methods to obtain public key of other party • Although public key can be used for confidentiality or authentication, asymmetric encryption algorithms are too slow • So usually want to use symmetric encryption to protect message contents • Can use asymmetric encryption to set up a session key

11. Simple Secret Key Distribution • Proposed by Merkle in 1979 • A generates a new temporary public key pair • A sends B the public key and A’s identity • B generates a session key Ks and sends encrypted Ks (using A’s public key) to A • A decrypts message to recover Ks and both use

12. Problem with Simple Secret Key Distribution • An adversary can intercept and impersonate both parties of protocol • A generates a new temporary public key pair {KUa, KRa} and sends KUa || IDa to B • Adversary E intercepts this message and sends KUe || IDa to B • B generates a session key Ks and sends encrypted Ks (using E’s public key) • E intercepts message, recovers Ks and sends encrypted Ks (using A’s public key) to A • A decrypts message to recover Ks and both A and B unaware of existence of E

13. Distribute Secret KeysUsing Asymmetric Encryption • if A and B have securely exchanged public-keys

14. Diffie-Hellman Key Exchange • First public-key type scheme proposed • By Diffie and Hellman in 1976 along with advent of public key concepts • A practical method for public exchange of secret key • Used in a number of commercial products

15. Diffie-Hellman Key Exchange • Use to set up a secret key that can be used for symmetric encryption • cannot be used to exchange an arbitrary message • Value of key depends on the participants (and their private and public key information) • Based on exponentiation in a finite (Galois) field (modulo a prime or a polynomial) - easy • Security relies on the difficulty of computing discrete logarithms (similar to factoring) – hard

16. Primitive Roots • From Euler’s theorem: aø(n) mod n=1 • Consider am mod n=1, GCD(a,n)=1 • must exist for m= ø(n) but may be smaller • once powers reach m, cycle will repeat • If smallest is m= ø(n) then a is called a primitive root • if p is prime, then successive powers of a “generate” the group mod p • Not every integer has primitive roots

17. Primitive Root Example: Power of Integers Modulo 19

18. Discrete Logarithms • Inverse problem to exponentiation is to find the discrete logarithm of a number modulo p • Namely find x where ax = b mod p • Written as x=loga b mod p or x=inda,p(b) • If a is a primitive root then always exists, otherwise may not • 3x = 4 mod 13 has no answer • 2x = 3 mod 13 has an answer 4 • while exponentiation is relatively easy, finding discrete logarithms is generally a hard problem

19. Diffie-Hellman Setup • All users agree on global parameters • large prime integer or polynomial q • α a primitive root mod q • Each user (eg. A) generates its key • choose a secret key (number): xA < q • compute its public key: yA = αxA mod q • Each user publishes its public key

20. Diffie-Hellman Key Exchange • Shared session key for users A and B is KAB: KAB = αxA.xB mod q = yAxB mod q (which B can compute) = yBxA mod q (which A can compute) • KAB is used as session key in symmetric encryption scheme between A and B • Attacker needs xA or xB, which requires solving discrete log

21. Diffie-Hellman Example • Given Alice and Bob who wish to swap keys • Agree on prime q=353 and α=3 • Select random secret keys: • A chooses xA=97, B chooses xB=233 • Compute public keys: • yA=397 mod 353 = 40 (Alice) • yB=3233 mod 353 = 248 (Bob) • Compute shared session key as: KAB= yBxA mod 353 = 24897 = 160 (Alice) KAB= yAxB mod 353 = 40233 = 160 (Bob)

22. Next Class • Hashing functions • Message digests • Read Chapters 11 and 12