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

Cryptographic Security 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 Systems2

Cryptographic Security 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=g k mod 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 = g x mod p given y, g, and p). Dennis Kafura – CS5204 – Operating Systems3

Cryptographic Security Key Exchange Protocol Public information  A prime number, p  A generator, g Steps  Alice chooses a random number a and computes u=g a mod p and sends u to Bob.  Bob chooses a random number b and computes v=g b mod p and sends v to Alice.  Bob computes the key k = u b = (g a ) b mod p.  Alice computes the key k = v a = (g b ) a mod p.  (note: both Bob and Alice have k = (g ab ) mod p) Dennis Kafura – CS5204 – Operating Systems4

Cryptographic Security 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,  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 Systems5

Cryptographic Security Identity-based encryption Dennis Kafura – CS5204 – Operating Systems6 Private Key Generator Encrypted with as public key authenticate send private key Alice Bob

Cryptographic Security Identity-based Encryption Dennis Kafura – CS5204 – Operating Systems7 SetupkExtract master-key Private Key Generator (PKG) Receiver Decrypt params Sender Encrypt M C M d ID ID

Cryptographic Security 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 Systems8

Cryptographic Security Identity-based encryption BasicIdent algorithms Setup Dennis Kafura – CS5204 – Operating Systems9

Cryptographic Security Identity-based Encryption Extract Encrypt Decrypt Dennis Kafura – CS5204 – Operating Systems10

Cryptographic Security 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 Systems11

Cryptographic Security Extensions Dennis Kafura – CS5204 – Operating Systems12 bilinear groups threshold secret sharing access tree ID-based attribute/fuzzy IDkey/policy-based