Security ranking among assumptions

Over the past decade bilinear maps have been used to build a large variety of cryptosystems. In parallel to new functionalities, we have also seen the emergence of many security assumptions. This leads to the general question of comparing two such assumptions. Boneh, Boyen and Goh introduced the Uber assumption as an attempt to offer a general framework for security assessment. Their idea is to propose a generic security assumption that can be specialized to suit the needs of any proof of protocols involving bilinear pairing. Even though the Uber assumption has been only stated in the bilinear setting, it can be easily restated to deal with ordinary Diffie-Hellman groups and assess other type of protocols. In this article, we explore some particular instances of the Uber assumption; namely the n-CDH-assumption, the n-CDH-assumption and the Q-CDH-assumption. We analyse the relationships between those assumption and more precisely from a security point of view. Our analysis does not rely on any special property of the considered group(s) and does not use the generic group model.