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CS283 Lecture 3 - Part 1 - Public Key Cryptography - 20090922

CS283 Lecture 3 - Part 1 - Public Key Cryptography - 20090922

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GWU CS 172/283 Autumn 2009 Public Key Cryptography GWU CS 172/283 - Autumn 2009 Holmblad - Lecture 03 – Part 1- Rev 20090922
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GWU CS 172/283 - Autumn 2009 Holmblad - Lecture 03 – Part 1- Rev 20090922 2 Lecture Topics - Polynomial Time - Diffie-Hellman Key Exchange - Discrete Log Problem - Public Key Cryptography - Rivest Shamir Adelman (RSA) Algorithm - Identity Based Encryption - Cryptographic Hash
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GWU CS 172/283 - Autumn 2009 Holmblad - Lecture 03 – Part 1- Rev 20090922 3 Mathematics Break – Polynomial Time (from Wikipedia) polynomial time refers to the running time of an algorithm , that is, the number of computation steps a computer or an abstract machine requires to evaluate the algorithm. An algorithm is said to be polynomial time if its running time is upper bounded by a polynomial in the size of the input for the algorithm. A Formal definition More formally, let T(n) be the running time of the algorithm on inputs of size at most n . Then the algorithm is polynomial time if there exists a polynomial p(n) such that, for all input sizes n , the running time T(n) is no larger than p(n)
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Diffie-Hellman Key Exchange GWU CS 172/283 - Autumn 2009 Holmblad - Lecture 03 – Part 1- Rev 20090922 4
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GWU CS 172/283 - Autumn 2009 Holmblad - Lecture 03 – Part 1- Rev 20090922 5 Diffie-Hellman Key Exchange Published in 1976 A Protocol for exchanging a secret key over a public channel. Select global parameters p , n and α . p is prime and α is of order n in Z p * . These parameters are public and known to all.
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GWU CS 172/283 - Autumn 2009 Holmblad - Lecture 03 – Part 1- Rev 20090922 6 Diffie-Hellman Key Exchange contd. Alice privately selects random b (secret) and sends to Bob α b mod p. Bob privately selects random c (secret) and sends to Alice α c mod p. Alice and Bob privately compute α bc mod p which is their shared secret . An observer Oscar can compute α bc if he knows either c or b or can solve the discrete log problem. This is a key agreement protocol .
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GWU CS 172/283 - Autumn 2009 Holmblad - Lecture 03 – Part 1- Rev 20090922 7 The cryptographic Strength of Diffie-Hellman is based on the difficulty of solving the Discrete Logarithm problem Given a multiplicative group G, an element γ G such that o( γ ) = n, and an element α < γ > Find the unique integer x, 0 x n-1 such that α = γ x x denoted as log γ α Not known to be solvable in polynomial time , however exponentiation is.
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GWU CS 172/283 - Autumn 2009 Holmblad - Lecture 03 – Part 1- Rev 20090922 8 Man in the Middle attack on Diffie- Hellman Diffie-Hellman key exchange is susceptible to a man-in-the- middle attack. Mallory captures b and c in transmission and replaces with own b’ and c’. Essentially runs two Diffie-Hellman’s. One with Alice and one with Bob.
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Public-Key Cryptography GWU CS 172/283 - Autumn 2009 Holmblad - Lecture 03 – Part 1- Rev 20090922 9
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GWU CS 172/283 - Autumn 2009 Holmblad - Lecture 03 – Part 1- Rev 20090922 10 Whitfield Diffie and Martin Hellman proposed Public Key Cryptography Computationally easy to encrypt/decrypt given access to the relevant key Computationally infeasible to derive the private key from the public key
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