Discrete logarithm and Diffie-Hellman problems in identity black-box groups

TL;DR

This paper explores the complexity of discrete logarithm and Diffie-Hellman problems in specific identity black-box groups, revealing cases where decisional and computational problems differ significantly in difficulty.

Contribution

It introduces a new class of groups G_{p,t} and analyzes their cryptographic problem complexities, showing contrasting hardness results for different problems.

Findings

Decisional Diffie-Hellman is solvable in polynomial time in G_{p,1}

Computational problems have query complexity Omega(p)

Quantum query complexity is Omega(sqrt(p)) for most problems

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.