On the Notion of Pseudo-Free Groups

We explore the notion of a pseudo-free group, first introduced by Hohenberger [Hoh03], and provide an alternative stronger definition. We show that if Z \(^{\rm *}_{n}\) is a pseudo-free abelian group (as we conjecture), then Z \(^{\rm *}_{n}\) also satisfies the Strong RSA Assumption [FO97,CS00,BP97]. Being a “pseudo-free abelian group” may be the strongest natural cryptographic assumption one can make about a group such as Z \(_{n}^{\rm *}\). More generally, we show that a pseudo-free group satisfies several standard cryptographic assumptions, such as the difficulty of computing discrete logarithms.