Adaptive Pseudofree Groups and Their Applications

In this paper we explore a powerful extension of the notion of pseudo-free groups, proposed by Rivest at TCC 2004. We identify, motivate, and study pseudo-freeness in face of adaptive adversaries who may learn solutions to other non-trivial equations before having to solve a new non-trivial equation. We present a novel, carefully crafted definition of adaptive pseudo-freeness that walks a fine line between being too weak and being unsatisfiable. We show that groups that satisfy our definition yield, via a generic construction, digital and network coding signature schemes. Finally, we obtain concrete constructions of such schemes in the RSA group by showing this group to be adaptive pseudo-free. In particular, we demonstrate the generality of our framework for signatures by showing that most existing schemes are instantiations of our generic construction.