Discrete logarithm and Diffie-Hellman problems in identity black-box groups
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We investigate the computational complexity of the discrete logarithm, the computational Diffie-Hellman and the decisional Diffie-Hellman problems in some identity black-box groups G_p,t, where p is a prime number and t is a positive integer. These are defined as quotient groups of vector space Z_p^t+1 by a hyperplane H given through an identity oracle. While in general black-box groups with unique encoding these computational problems are classically all hard and quantumly all easy, we find that in the groups G_p,t the situation is more contrasted. We prove that while there is a polynomial time probabilistic algorithm to solve the decisional Diffie-Hellman problem in G_p,1, the probabilistic query complexity of all the other problems is Omega(p), and their quantum query complexity is Omega(sqrt(p)). Our results therefore provide a new example of a group where the computational and the decisional Diffie-Hellman problems have widely different complexity.
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