We present a general framework for constructing non-interactive universally composable (UC) commitment schemes that are secure against adaptive adversaries in the non-erasure model under a reusable common reference string. Previously, such "fully-equipped" UC commitment schemes have been known only in [5, 6], with strict expansion factor O(κ); meaning that to commit A bits, communication strictly requires O(λκ) bits, where κ denotes the security parameter. Efficient construction of a fully-equipped UC commitment scheme is a long-standing open problem. We introduce new abstraction, called all-but-many encryption (ABME), and prove that it captures a fully-equipped UC commitment scheme. We propose the first fully-equipped UC commitment scheme with optimal expansion factor Ω(1) from our ABME scheme related to the DCR assumption. We also provide an all-but-many lossy trapdoor function (ABM-LTF) from our DCR-based ABME scheme, with a better lossy rate than.
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