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

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

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ublic Key Cryptography Public Key Cryptography GWU CS 172/283 Autumn 2009 GWU CS 172/283 - Autumn 2009 Holmblad - Lecture 03 – Part 1- Rev 20090922

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Lecture Topics - Polynomial Time - Diffie-Hellman Key Exchange - Discrete Log Problem - ublic Key Cryptography Public Key Cryptography - Rivest Shamir Adelman (RSA) Algorithm - Identity Based Encryption - Cryptographic Hash GWU CS 172/283 - Autumn 2009 Holmblad - Lecture 03 – Part 1- Rev 20090922 2
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. Formal definition 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) GWU CS 172/283 - Autumn 2009 Holmblad - Lecture 03 – Part 1- Rev 20090922 3

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iffie- ellman Key Exchange Diffie Hellman Key Exchange GWU CS 172/283 - Autumn 2009 Holmblad - Lecture 03 – Part 1- Rev 20090922 4
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 . is prime and of order n in Z * These parameters p is prime and α is of order n in Z p . These parameters are public and known to all. GWU CS 172/283 - Autumn 2009 Holmblad - Lecture 03 – Part 1- Rev 20090922 5

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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 . n observer Oscar can compute c he knows either c or 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 . GWU CS 172/283 - Autumn 2009 Holmblad - Lecture 03 – Part 1- Rev 20090922 6
The cryptographic Strength of Diffie-Hellman based on the difficulty of solving the 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 denoted as log x denoted as log γ α • Not known to be solvable in polynomial time , however exponentiation is. GWU CS 172/283 - Autumn 2009 Holmblad - Lecture 03 – Part 1- Rev 20090922 7

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an in the Middle attack on Diffie 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 places with own b’ and c’ replaces with own b and c .
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CS283 Lecture 3 - Part 1 - Public Key Cryptography - 20090922

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