We propose a general cryptographic primitive called lossy trapdoor functions (lossy TDFs), and use it to develop new approaches for constructing several important cryptographic tools, including (injective) trapdoor functions, collision-resistant hash functions, oblivious transfer, and chosen ciphertext-secure cryptosystems (in the standard model). All of these constructions are simple, efficient, and black-box. We realize lossy TDFs based on a variety of cryptographic assumptions, including the hardness of the decisional Diffie-Hellman (DDH) problem, and the hardness of the “learning with errors ” problem (which is implied by the worst-case hardness of various lattice problems). Taken together, our results resolve some long-standing open problems in cryptography. They give the first injective trapdoor functions based on problems not directly related to integer factorization, and provide the first chosen ciphertext-secure cryptosystem based solely on worst-case complexity assumptions. A preliminary version of this work appeared in the 40th ACM Symposium on Theory of Computing (STOC 2008). A majority of this work was performed while at SRI International. This material is based upon work supported by the Nationa
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