Paper 2006/178

A New Cryptosystem Based On Hidden Order Groups

Amitabh Saxena and Ben Soh


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.

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

Available format(s)
Public-key cryptography
Publication info
Published elsewhere. arXiv report archive (eprint arXiv:cs/0605003)
Contact author(s)
asaxena @ cs latrobe edu au
2007-03-05: last of 6 revisions
2006-05-30: received
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      author = {Amitabh Saxena and Ben Soh},
      title = {A New Cryptosystem Based On Hidden Order Groups},
      howpublished = {Cryptology ePrint Archive, Paper 2006/178},
      year = {2006},
      note = {\url{}},
      url = {}
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