Cyrtographic Security

1. Cyrtographic Security Identity-based Encryption Dennis Kafura – CS5204 – Operating Systems

2. Diffie-Hellman Key Exchange How can two parties come to possess a shared secret using only insecure channels of communication? Assumes passive eavesdropping only (i.e. susceptible to active (wo)man-in-the-middle attack) Relies on prime number groups (more later) Same/similar techniques underlie more recent cryptographic methods Dennis Kafura – CS5204 – Operating Systems

3. Diffie-Hellman Key Exchange • Some mathematics • If p is prime number, then the numbers 1..p-1 form a group of order p-1 with multiplication modulo p as its operator. • A generator, g, is any number 1..p-1 such that for all n in 1..p-1 there is a power k such that n=gkmod p. • Example: 3 is a generator for the group with p=7 • Notation: • Operations: • Security based on computational infeasibility of solving the discrete logarithm problem (i.e., finding x if y = gx mod p given y, g, and p). Dennis Kafura – CS5204 – Operating Systems

4. Key Exchange Protocol • Public information • A prime number, p • A generator, g • Steps • Alice chooses a random number a and computes u=ga mod p and sends u to Bob. • Bob chooses a random number b and computes v=gb mod p and sends v to Alice. • Bob computes the key k = ub = (ga)b mod p. • Alice computes the key k = va = (gb)a mod p. • (note: both Bob and Alice have k = (gab) mod p) Dennis Kafura – CS5204 – Operating Systems

5. Identity-based encryption • Public-key encryption • Identity is conveyed in a certificate from a certificate authority that binds the public key to the identity • Certificate must be obtained in advance • Certificate authority is trusted to validate claim of identity • Identity-based encryption • Identity itself serves as the public key (e.g, bob@company.com) • No advance preparation needed • Trusted service validates claim of identity • Key escrow issue (trusted service can recreate secret key associated with an identity) Dennis Kafura – CS5204 – Operating Systems

6. Identity-based encryption Bob Alice send private key Encrypted with bob@company.com as public key Private Key Generator authenticate bob@company.com Dennis Kafura – CS5204 – Operating Systems

7. Identity-based Encryption Private Key Generator (PKG) master-key Extract Setup k ID ID params Receiver dID Decrypt C Encrypt M M Sender Dennis Kafura – CS5204 – Operating Systems

8. Bilinear Maps • Some mathematics • Fortunately, groups with these properties can be generated algorithmically using a positive integer seed value (security parameter) k. Dennis Kafura – CS5204 – Operating Systems

9. Identity-based encryption BasicIdent algorithms Setup Dennis Kafura – CS5204 – Operating Systems

10. Identity-based Encryption Extract Encrypt Decrypt Dennis Kafura – CS5204 – Operating Systems

11. Why does this work? Encryption bitwise exclusive-ors M with: Decryption bitwise exclusive-ors V with: These masks are the same since: Dennis Kafura – CS5204 – Operating Systems

12. Extensions bilinear groups ID-based threshold secret sharing attribute/fuzzy ID access tree key/policy-based Dennis Kafura – CS5204 – Operating Systems