Introduction to Pubic Key Encryption CSCI 5857: Encoding and Encryption

Outline Basic concepts and goals of public key encryption One-way functions Trapdoor functions

Public/Private Key Pairs Recipient (Alice) generates key pair: – Public key k PU Does not have to be kept secret Distributed to all senders (such as Bob) – Private key k PR Kept secret by Alice

Public Key Encryption Bob uses Alice’s public key k PU to encrypt message – C = E(k PU, P) Alice uses her private key k PR to decrypt message – P = D(k PR, C)

Public Key Security Central idea: Adversary cannot determine private key from corresponding public key – Could theoretically find private key, but computationally infeasible to do so – Cannot read intercepted messages encrypted with public key

Public and Symmetric Keys Problem: How to securely distribute a symmetric key K S ? Solution: 1.Use public key encryption to securely send it 2.Use faster symmetric key algorithm (like AES) to securely transmit the rest of the message

Public Key Math Public key algorithms are mathematical functions of integer numbers – Keys are large numbers (hundreds of digits long) – Plaintext translated to large numbers (not bits) – Encryption is a mathematical function of plaintext and key which creates another large number as ciphertext

One-Way Functions One-way functions: – Function: y = f (x) – Inverse function: x = f -1 (y) Given x, y = f (x) very easy to compute Given y, x = f -1 (y) computationally infeasible to compute Example: Factoring – p and q are very large prime numbers – n = p x q is easy to compute – Factoring n into p and q infeasible Must try almost all possible p and q

9 Trapdoor One-Way Functions Trapdoor functions: Given one-way function: y = f (x) There exists some “secret trapdoor” that allows x = f -1 (y) to be easily computed Example (very simple): n = p x q product of two large primes Factoring n into p and q to find p infeasible Finding p is easy if know q – q is a “trapdoor” for finding p from n

Trapdoor Functions and Private Keys Idea behind public-key encryption: Encryption function C = E (K PU, P) must be one way – Must not be able to compute P from C Must have trapdoor to allow decryption – Must be able to easily compute P from C if know trapdoor Trapdoor = private key

Types of Trapdoor One-Way Functions Discrete Logarithms – RSA, Rabin, ElGamal, Diffie-Hellman – Easy to implement, well understood Elliptic Curve – Discrete logarithms represented as curves – Much faster than discrete logarithms NP-Complete problems – Example: “knapsack problem”, Merkle and Hellman (1978) – Exponential time to solve problem – Easy to confirm solution if given

What’s Next Let me know if you have any questions Continue on to the next lecture on RSA Public Key Encryption