CSCE 715: Network Systems Security Chin-Tser Huang University of South Carolina

09/14/20102 Key Management Asymmetric encryption helps address key distribution problems Two aspects distribution of public keys use of public-key encryption to distribute secret keys

09/14/20103 Distribution of Public Keys Four alternatives of public key distribution Public announcement Publicly available directory Public-key authority Public-key certificates

09/14/20104 Public Announcement Users distribute public keys to recipients or broadcast to community at large E.g. append PGP keys to messages or post to news groups or list Major weakness is forgery anyone can create a key claiming to be someone else’s and broadcast it can masquerade as claimed user before forgery is discovered

09/14/20105 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

09/14/20106 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

09/14/20107 Public-Key Authority

09/14/20108 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

09/14/20109 Public-Key Certificates

09/14/ Distribute Secret Keys Using 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

09/14/ 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 K s and sends encrypted K s (using A’s public key) to A A decrypts message to recover K s and both use

09/14/ Problem with Simple Secret Key Distribution An adversary can intercept and impersonate both parties of protocol A generates a new temporary public key pair {KU a, KR a } and sends KU a || ID a to B Adversary E intercepts this message and sends KU e || ID a to B B generates a session key K s and sends encrypted K s (using E’s public key) E intercepts message, recovers K s and sends encrypted K s (using A’s public key) to A A decrypts message to recover K s and both A and B unaware of existence of E

09/14/ Distribute Secret Keys Using Asymmetric Encryption if A and B have securely exchanged public-keys ?

09/14/ Problem with Previous Scenario Message (4) is not protected by N 2 An adversary can intercept message (4) and replay an old message or insert a fabricated message

09/14/ Order of Encryption Matters What can be wrong with the following protocol? A  B:N B  A:E KUa [E KRb [K s ||N]] An adversary sitting between A and B can get a copy of secret key K s without being caught by A and B! Reverse the order of encryption using KR b and encryption using KU a can avoid this attack

09/14/ Diffie-Hellman Key Exchange First publicly proposed public-key type scheme 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

09/14/ 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

09/14/ Primitive Roots From Euler’s theorem: a ø(n) mod n=1 Consider a m 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 and a is a primitive root of p, then successive powers of a “generate” the group mod p Not every integer has primitive roots

09/14/ Primitive Root Example: Power of Integers Modulo 19

09/14/ Discrete Logarithms Inverse problem to exponentiation is to find the discrete logarithm of a number modulo p Namely find x where a x = b mod p Written as x=log a b mod p or x=dlog a,p (b) If a is a primitive root of p then discrete logarithm always exists, otherwise may not 3 x = 4 mod 13 has no answer 2 x = 3 mod 13 has an answer 4 While exponentiation is relatively easy, finding discrete logarithms is generally a hard problem

09/14/ Diffie-Hellman Setup All users agree on global parameters large prime integer or polynomial q α which is a primitive root mod q Each user (e.g. A) generates its key choose a private key (number): x A < q compute its public key: y A = α x A mod q Each user publishes its public key

09/14/ Diffie-Hellman Key Exchange Shared session key for users A and B is K AB : K AB = α x A. x B mod q = y A x B mod q (which B can compute) = y B x A mod q (which A can compute) K AB is used as session key in symmetric encryption scheme between A and B Attacker needs x A or x B, which requires solving discrete log

09/14/ 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 x A =97, B chooses x B =233 Compute public keys: y A =3 97 mod 353 = 40 (Alice) y B =3 233 mod 353 = 248 (Bob) Compute shared session key as: K AB = y B x A mod 353 = = 160 (Alice) K AB = y A x B mod 353 = = 160 (Bob)

09/14/ Elliptic Curve Cryptography Majority of public-key crypto (RSA, D-H) use either integer or polynomial arithmetic with very large numbers/polynomials Imposes a significant load in storing and processing keys and messages An alternative is to use elliptic curves Offers same security with smaller bit sizes

09/14/ Real Elliptic Curves An elliptic curve is defined by an equation in two variables x and y, with coefficients Consider a cubic elliptic curve of form y 2 = x 3 + ax + b where x, y, a, b are all real numbers also define zero point O Have addition operation for elliptic curve geometrically, sum of P+Q is reflection of intersection R

09/14/ Real Elliptic Curve Example

09/14/ Finite Elliptic Curves Elliptic curve cryptography uses curves whose variables and coefficients are finite Two families are commonly used prime curves E p (a,b) defined over Z p use integers modulo a prime best in software binary curves E 2 m (a,b) defined over GF(2 m ) use polynomials with binary coefficients best in hardware

09/14/ Elliptic Curve Cryptography ECC addition is analog of modulo multiply ECC repeated addition is analog of modulo exponentiation Need a “hard” problem equivalent to discrete logarithm Q=kP, where Q, P belong to a prime curve is “easy” to compute Q given k, P but “hard” to find k given Q, P known as the elliptic curve logarithm problem Certicom example: E 23 (9,17)

09/14/ ECC Diffie-Hellman Can do key exchange analogous to D-H Users select a suitable curve E p (a,b) Select base point G=(x 1, y 1 ) with large order n s.t. nG=O A and B select private keys n A <n, n B <n Compute public keys: P A =n A ×G, P B =n B ×G Compute shared key: K=n A ×P B, K=n B ×P A same since K=n A ×n B ×G

09/14/ ECC Encryption/Decryption Must first encode any message M as a point on the elliptic curve P m Select suitable curve and point G as in D-H Each user chooses private key n A <n and computes public key P A =n A ×G To encrypt P m : C m ={kG, P m +kP B }, k random To decrypt C m : P m +kP B –n B (kG) = P m +k(n B G)–n B (kG) = P m

09/14/ ECC Security Relies on elliptic curve logarithm problem Fastest method is “Pollard rho method” Compared to factoring, ECC can use much smaller key sizes than with RSA For equivalent key lengths computations are roughly equivalent Hence for similar security ECC offers significant computational advantages

09/14/ Comparable Key Sizes 1

09/14/ Next Class Message authentication Hashing functions Message digests Read Chapters 11 and 12