El Gamal and Diffie Hellman

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Presentation transcript:

El Gamal and Diffie Hellman ElGamal Cryptosystem In Practice Diffie-Hellman El Gamal and Diffie Hellman CSCI284, 162 Spring 2008 GWU

The ElGamal Cryptosystem is based on the Discrete Log problem: Given a multiplicative group G, an element  G such that o() = n, and an element  <> Find the unique integer x, 0  x  n-1 such that  = x x denoted as log Not known to be doable in polynomial time, however exponentiation is. Hence DL is a possible one-way function 2/22/2019 CS284-162/Spring08/GWU/Vora/Discrete Log

CS284-162/Spring08/GWU/Vora/Discrete Log El Gamal Cryptosystem Let p a prime such that DL in Zp* is infeasible Let  Zp* be a primitive element P = Zp* C = Zp* X Zp* and K = {(p, , a, ): =a (mod p)} public key = (p, , ) and private key = a For a secret random number k Zp-1 eK(x, k) = (y1, y2) y1 = k mod p y1 = xk mod p dK (y1, y2) = y2( y1a)-1 mod p 2/22/2019 CS284-162/Spring08/GWU/Vora/Discrete Log

CS284-162/Spring08/GWU/Vora/Discrete Log Example p = 2579  = 2 a = 1391 Encrypt message: 2079 2/22/2019 CS284-162/Spring08/GWU/Vora/Discrete Log

CS284-162/Spring08/GWU/Vora/Discrete Log Practicalities More efficient attacks possible unless elliptic curve DL, for which these efficient attacks are not known. Modulus required for security: 2160 with elliptic curves 21880 without DL over elliptic curves very hot problem. 2/22/2019 CS284-162/Spring08/GWU/Vora/Discrete Log

Diffie-Hellman Key Exchange Protocol for exchanging secret key over public channel. Select global parameters p, n and . p is prime and  is of order n in Zp*. These parameters are public and known to all. 2/22/2019 CS284-162/Spring08/GWU/Vora/Discrete Log

Diffie-Hellman Key Exchange contd. Alice privately selects random b and sends to Bob b mod p. Bob privately selects random c and sends to Alice c mod p. Alice and Bob privately compute bc mod p which is their shared secret. An observer Oscar can compute bc if he knows either c or b or can solve the discrete log problem. This is a key agreement protocol. 2/22/2019 CS284-162/Spring08/GWU/Vora/Discrete Log

Diffie-Hellman problem Given a multiplicative group G, an element G of order n and two elements ,   <> Computational Diffie-Hellman: Find  such that log   log   log (mod n) Equivalently, given b, and c find bc Decision Diffie-Hellman Given an additional   <> Determine if log   log   log (mod n) Equivalently, given b, c, and d determine if d  bc (mod n) 2/22/2019 CS284-162/Spring08/GWU/Vora/Discrete Log

CS284-162/Spring08/GWU/Vora/Discrete Log An attack 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. 2/22/2019 CS284-162/Spring08/GWU/Vora/Discrete Log