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Published byElfrieda Wilkinson Modified over 8 years ago
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Introduction to Pubic Key Encryption CSCI 5857: Encoding and Encryption
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Outline Basic concepts and goals of public key encryption One-way functions Trapdoor functions
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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
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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)
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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
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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
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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
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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
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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
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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
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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
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What’s Next Let me know if you have any questions Continue on to the next lecture on RSA Public Key Encryption
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