Lossy trapdoor functions (LTFs), introduced by Peiker and Waters in STOC'08, are functions that may be working in another injective mode or a lossy mode. Given such a function key, it is impossible to distinguish an injective key from a lossy key for any (probabilistic) polynomial-time adversary. This paper studies lossy trapdoor functions with tight security. First, we give a formal definition for tightly secure LTFs. Loosely speaking, a collection of LTFs is tightly secure if the advantage to

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... distinguish a tuple of injective keys from a tuple of lossy keys does not degrade in the number of function keys. Then, we show that tightly secure LTFs can be used to construct public-key encryption schemes with tight CPA security in a multiuser, multichallenge setting, and with tight CCA security in a multiuser, one-challenge setting. Finally, we present a construction of tightly secure LTFs from the decisional Diffie-Hellman assumption.