In this paper, we propose the notion of n-evasive all-but-many lossy trapdoor functions (ABM-LTFs), which is an extended abstraction of all-but-n lossy trapdoor functions (ABN-LTFs) proposed by Hemenway et al. in Asiacrypt 2011 and, at the same time, is a special case of ABM-LTFs proposed by Hofheinz in Eurocrypt 2012. We show two constructions of n-evasive ABM-LTFs. The first one is based on the decisional composite residuosity (DCR) assumption, and the second one is from chameleon hash functions and ABN-LTFs. Both of the constructions are based on reasonable assumptions with tight security reductions. Similar to ABN-LTFs and ABM-LTFs, n-evasive ABM-LTFs can be employed to construct indistinguishability-based selective opening chosen-ciphertext secure public-key encryption (PKE) schemes and may have other applications in cryptography. The instantiation of n-evasive ABM-LTFs can be based on the well-known assumption, for example, DCR, and the security reduction is much tighter than that of ABM-LTFs. On the other hand, the black-box PKE construction from n-evasive ABM-LTFs is more general than that from ABN-LTFs. Copyright © 2014 John Wiley & Sons, Ltd.