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.