(The) $l$-th power diffie-hellman problem and the $l$-th root diffie-hellman problem

Abstract

There are many variants of the computational Diffie-Hellman problem that are necessary to provide security of many cryptographic schemes. Two of them are the square Diffie-Hellman problem and the square root Diffie-Hellman problem. Dongyoung Roh and Sang Geun Hahn proved that these two problems are polynomial-time equivalent under a certain condition [17]. In this paper, we generalize this result. We introduce the $l$-th power Diffie-Hellman problem and the $l$-th root Diffie-Hellman problem and show that these two problems are polynomial-time equivalent for $l=O(\log p)$ under a condition similar to that of [17], where $p$ is the order of the underlying group.