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

*Amitabh Saxena and Ben Soh*

**Abstract: **
Let $G_1$ be a cyclic multiplicative group of order $n$. It is known that the Diffie-Hellman problem is random self-reducible in $G_1$ with respect to a fixed generator $g$ if $\phi(n)$ is known. That is, given $g, g^x\in G_1$ and having oracle access to a `Diffie-Hellman Problem' solver with fixed generator $g$, it is possible to compute $g^{1/x} \in G_1$ in polynomial time. On the other hand, it is not known if such a reduction exists when $\phi(n)$ 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.

**Category / Keywords: **public-key cryptography /

**Date: **received 22 Feb 2006, last revised 27 Feb 2006, withdrawn 27 Feb 2006

**Contact author: **amitabh123 at gmail com

**Available format(s): **(-- withdrawn --)

**Version: **20060227:165351 (All versions of this report)

**Short URL: **ia.cr/2006/076

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