Canetti et al. [7] recently proposed a new framework - termed Generalized Universal Composability (GUC) - for properly analyzing concurrent execution of cryptographic protocols in the presence of a global setup, and constructed the first known GUC-secure implementations of commitment (GUCC) and zero-knowledge (GUC ZK), which suffice to implement any two-party or multi-party functionality under several natural and relatively mild setup assumptions. Unfortunately, the feasibility results of [7] used rather inefficient constructions. In this paper, we dramatically improve the efficiency of (adaptively-secure) GUCC and GUC ZK assuming data erasures are allowed. Namely, using the same minimal setup assumptions as those used by [7], we build (1) a direct and efficient constant-round GUC ZK for R from any "dense" Ω-protocol [21] for R. As a corollary, we get a semi-efficient construction from any Σ-protocol for R (without doing the Cook-Levin reduction), and a very efficient GUC ZK for proving knowledge of a discrete log representation. (2) the first constant-rate (and constant-round) GUCC scheme. Additionally, we show how to properly model a random oracle in the GUC framework without losing deniability, which is one of the attractive features of the GUC framework. In particular, by adding the random oracle to the setup assumptions used by [7], we build the first two-round (which we show is optimal), deniable, straight-line extractable and simulatable ZK proof for any NP relation R.
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