Paper 2006/076

A Cryptosystem Based on Hidden Order Groups and Its Applications in Highly Dynamic Group Key Agreement

Amitabh Saxena and Ben Soh

Abstract

Let $$ be a cyclic multiplicative group of order $$. It is known that the Diffie-Hellman problem is random self-reducible in $$ with respect to a fixed generator $$ if $$ is known. That is, given $$ and having oracle access to a `Diffie-Hellman Problem' solver with fixed generator $$, it is possible to compute $$ in polynomial time. On the other hand, it is not known if such a reduction exists when $$ is unknown. We exploit this ``gap'' to construct a cryptosystem based on hidden order groups by presenting a practical implementation of a novel cryptographic primitive called \emph{Strong Associative One-Way Function} (SAOWF). SAOWFs have interesting applications like one-round group key agreement. We demonstrate this by presenting an efficient group key agreement protocol for dynamic ad-hoc groups. Our cryptosystem can be considered as a combination of the Diffie-Hellman and RSA cryptosystems.