Multi-use unidirectional proxy re-signatures

In 1998, Blaze, Bleumer, and Strauss suggested a cryptographic primitive termed proxy re-signature in which a proxy transforms a signature computed under Alice's secret key into one from Bob on the same message. The proxy is only semi-trusted in that it cannot learn any signing key or sign arbitrary messages on behalf of Alice or Bob. At CCS 2005, Ateniese and Hohenberger revisited this primitive by providing appropriate security definitions and efficient constructions in the random oracle model. Nonetheless, they left open the problem of constructing a multi-use unidirectional scheme where the proxy is only able to translate in one direction and signatures can be re-translated several times. This paper provides the first steps towards efficiently solving this problem, suggested for the first time 10 years ago, and presents the first multi-hop unidirectional proxy re-signature schemes. Although our proposals feature a linear signature size in the number of translations, they are the first multi-use realizations of the primitive that satisfy the requirements of the Ateniese-Hohenberger security model. The first scheme is secure in the random oracle model. Using the same underlying idea, it readily extends into a secure construction in the standard model (i.e. the security proof of which avoids resorting to the random oracle idealization). Both schemes are computationally efficient but require newly defined Diffie-Hellman-like assumptions in bilinear groups.