lec9-4 - Diffie Hellman Elgamal Discrete Logarithm Problem...

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Diffie, Hellman, Elgamal Discrete Logarithm Problem G : cyclic group of order p g : a generator of G Discrete logarithm problem Find log g ( h ) with non-negligible probability, for a uniform random h G Solver’s performance measured by success probability 2 Discrete Logarithm Problem Discrete logarithm assumption DL problem is hard: any solver which spends reasonable amount of time has only negligible success probability 3 Diffie-Hellman 1976: “New Directions in Cryptography” Many of modern ideas of PKC introduced Public-key encryption, one way function, digital signature, ... Public-key based key exchange Securely share a symmetric key using only public channels 4

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Diffie & Hellman DH key exchange G : a cyclic group, with generator g Originally G = Z p * Alice: pick a and send g a to Bob Bob: pick b and send g b to Alice Both: compute g ab =( g a ) b =( g b ) a K = g ab : shared symmetric key 6 Security of DH First, some authentication necessary! Second, using g ab directly may not be good At least DL assumption necessary If a can be learned from g a , then g ab =( g b ) a In general, DL assumption not enough 7 Diffie-Hellman Problem Computational Diffie-Hellman problem: computing g xy , given g x and g y , for random x & y Computational Diffie-Hellman assumption: this is difficult on G 8
DL vs CDH CDH is easier: if DL can be solved then CDH can be too It is known that if G satisfies some mild conditions, CDH and DL are equivalent

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