Year: 2019
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Authors: Ivanyos, Gabor, Joux, Antoine, Santha, Miklos
Institutions: HUN-REN Institute for Computer Science and Control, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne Université, Helmholtz Center for Information Security, OURAGAN: Outils de Résolution Algébriques pour la Géométrie et ses Applications, Centre for Quantum Technologies, Centre National de la Recherche Scientifique, Université Paris Cité, Institut de Recherche en Informatique Fondamentale
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: G2973756291, G5428356290
OpenAlex ID: W4297746007