## Cryptology ePrint Archive: Report 2006/178

**A New Cryptosystem Based On Hidden Order Groups**

*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 (see theorem 3.2). On the other hand, it is not known if such a reduction exists when $\phi(n)$ is unknown (see conjuncture 3.1). We exploit this ``gap'' to construct a cryptosystem based on hidden order groups and present a practical implementation of a novel cryptographic primitive called an \emph{Oracle Strong Associative One-Way Function} (O-SAOWF). O-SAOWFs have applications in multiparty protocols. We demonstrate this by presenting a key agreement protocol for dynamic ad-hoc groups.

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

**Publication Info: **arXiv report archive (eprint arXiv:cs/0605003)

**Date: **received 19 May 2006, last revised 5 Mar 2007

**Contact author: **asaxena at cs latrobe edu au

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation

**Note: **Updates on 05/03/07: A few typos corrected.

**Version: **20070305:200052 (All versions of this report)

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