Key Management and Diffie-Hellman

1. INCS 741: Cryptography Key Management and Diffie-Hellman Dr. Monther Aldwairi New York Institute of Technology- Amman Campus 12/3/2009 Dr. Monther Aldwairi

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

3. Distribution of Public Keys • can be considered as using one of: • 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 • until forgery is discovered can masquerade as claimed user

5. Publicly Available Directory • Registering keys with a public directory • directory must be trusted entity or organization • contains {name, public-key} entries • is periodically published and accessed electronically • participants register securely with directory • participants can replace key at any time • still vulnerable to tampering or forgery • adversary succeeds in obtaining the private key of the directory authority • could tamper with the records kept by the authority.

6. Public-Key Authority • improve security by tightening control over distribution of keys from directory • has properties of directory • and requires users to know public key for the directory • then users interact with directory to obtain any desired public key securely • does require real-time access to directory when keys are needed

7. Public-Key Authority

8. Public-Key Authority Drawbacks • The public-key authority could be somewhat of a bottleneck in the system • A user must appeal to the authority for a public key for every other user that it wishes to contact. • As before, the directory of names and public keys maintained by the authority is vulnerable to tampering. Dr. Monther Aldwairi

9. 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, rights of use etc • with all contents signed by a trusted Public-Key or Certificate Authority (CA) • can be verified by anyone who knows the public-key certificate authorities public-key

10. Requirements • Any participant determine the name and public key of the certificate's owner. • Any participant can verify that the certificate originated from the certificate authority. • Only the certificate authority can create and update certificates. • Any participant can verify the currency of the certificate. Dr. Monther Aldwairi

11. Public-Key Certificates

12. Public-Key Distribution of Secret Keys • use previous methods to obtain public-key • can use for secrecy or authentication • but public-key algorithms are slow • so usually want to use private-key encryption to protect message contents • hence need a session key • have several alternatives for negotiating a suitable session

13. Simple Secret Key Distribution • proposed by Merkle in 1979 • A generates a new temporary public key pair • A sends B the public key and their identity • B generates a session key K sends it to A encrypted using the supplied public key • A decrypts the session key and both use (discard keys) • problem is that an opponent can intercept and impersonate both halves of protocol • adversary who can intercept messages and then either relay the intercepted message or substitute another message

14. Man in the Middle Attack • A generates a public/private key pair {PUa, PRa} and transmits a message intended for B consisting of PUa and an identifier of A, IDA. • E intercepts the message, creates its own public/private key pair {PUe, PRe} and transmits PUe||IDA to B. • B generates a secret key, Ks, and transmits E(PUe, Ks). • E intercepts the message, and learns Ks by computing D(PRe, E(PUe, Ks)). • E transmits E(PUa, Ks) to A. Dr. Monther Aldwairi

15. Public-Key Distribution of Secret Keys • if have securely exchanged public-keys:

16. Hybrid Key Distribution • retain use of private-key KDC • shares secret master key with each user • distributes session key using master key • public-key used to distribute master keys • especially useful with widely distributed users • rationale • performance • backward compatibility

17. Diffie-Hellman Key Exchange • first public-key type scheme proposed • by Diffie & Hellman in 1976 along with the exposition of public key concepts • note: now know that Williamson (UK CESG) secretly proposed the concept in 1970 • is a practical method for public exchange of a secret key • used in a number of commercial products

18. Diffie-Hellman Key Exchange • a public-key distribution scheme • cannot be used to exchange an arbitrary message • rather it can establish a common key • known only to the two participants • 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

19. Discrete Logarithms • a primitive root of a prime number p as one whose powers modulo p generate all the integers from 1 to p-1 • For any integer b and a primitive root a of prime number p, we can find a unique exponent i such that • b ≡ ai (mod p) where 0 ≤ i ≤(p-1) • The exponent i is referred to as the discrete logarithm of b for the base a, mod p. • dloga,p (b)= i Dr. Monther Aldwairi

20. Diffie-Hellman Setup • all users agree on global parameters: • large prime integer or polynomial q • α being a primitive root mod q • each user (eg. A) generates their key • chooses a secret key (number): xA < q • compute their public key: yA = α xA mod q • each user makes public that key yA

21. Diffie-Hellman Key Exchange • shared session key for users A & B is KAB: KAB = α xA.xB mod q = yAxB mod q (which B can compute) = yBxA mod q (which A can compute) • KAB is used as session key in private-key encryption scheme between Alice and Bob • if Alice and Bob subsequently communicate, they will have the same key as before, unless they choose new public-keys • attacker needs an x, must solve discrete log

22. Dr. Monther Aldwairi

23. Security of the Diffie-Hellman • An adversary only has the following ingredients to work with: q, α, YA, and YB. • XA and XB are private • Calculate discrete logarithm to determine the key • XB = dloga,q (YB) • The security of the Diffie-Hellman key exchange lies in the fact that, • it is relatively easy to calculate exponentials modulo a prime • it is very difficult to calculate discrete logarithms. For large primes, the latter task is considered infeasible. Dr. Monther Aldwairi

24. Diffie-Hellman Example • users Alice & 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 respective 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)

25. Attack Example • An attacker would have available the following information: • q = 353; α = 3; YA = 40; YB = 248 • By brute force, an attacker E can determine the common key by discovering a solution to eqns • 3a mod 353 = 40 • 3b mod 353 = 248 • calculate powers of 3 modulo 353, stopping when the result equals either 40 or 248 Dr. Monther Aldwairi

26. Key Exchange Protocols • users could create random private/public D-H keys each time they communicate • users could create generate a long-lasting private value Xi (for user i) and calculate a public value Yi. • These public values and q and α, are stored in some central directory • user j can access i's public value, calculate KAB • both of these are vulnerable to a meet-in-the-Middle Attack • authentication of the keys is needed