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 t...