Let G1 be a cyclic multiplicative group of order n. It is known that the
Diffie-Hellman problem is random self-reducible in G1 with respect to a
fixed generator g if ϕ(n) is known. That is, given g,gx∈G1 and
having oracle access to a `Diffie-Hellman Problem' solver with fixed generator
g, it is possible to compute g1/x∈G1 in polynomial time (see
theorem 3.2). On the other hand, it is not known if such a reduction exists
when ϕ(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.Comment: removed examples for multiparty key agreement and join protocols,
since they are redundan