Discrete logarithm and DiffieHellman problems in identity blackbox groups
Abstract
We investigate the computational complexity of the discrete logarithm, the computational DiffieHellman and the decisional DiffieHellman problems in some identity blackbox 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 blackbox 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 DiffieHellman 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 DiffieHellman problems have widely different complexity.
 Publication:

arXiv eprints
 Pub Date:
 November 2019
 arXiv:
 arXiv:1911.01662
 Bibcode:
 2019arXiv191101662I
 Keywords:

 Quantum Physics;
 Computer Science  Computational Complexity