Efficiency Preserving Transformations for Concurrent Non-malleable Zero Knowledge

Ever since the invention of Zero-Knowledge by Goldwasser, Micali, and Rackoff [1], Zero-Knowledge has become a central building block in cryptography - with numerous applications, ranging from electronic cash to digital signatures. The properties of Zero-Knowledge range from the most simple (and not particularly useful in practice) requirements, such as honest-verifier zero-knowledge to the most demanding (and most useful in applications) such as non-malleable and concurrent zero-knowledge. In this paper, we study the complexity of efficient zero-knowledge reductions, from the first type to the second type. More precisely, under a standard complexity assumption (ddh), on input a public-coin honest-verifier statistical zero knowledge argument of knowledge π′ for a language L we show a compiler that produces an argument system π for L that is concurrent non-malleable zero-knowledge (under non-adaptive inputs – which is the best one can hope to achieve [2,3]). If κ is the security parameter, the overhead of our compiler is as follows: The round complexity of π is $r+\tilde{O}(\log\kappa)$ rounds, where r is the round complexity of π′. The new prover $\mathcal{P}$ (resp., the new verifier $\mathcal{V}$) incurs an additional overhead of (at most) $r+{\kappa\cdot\tilde{O}(\log^2\kappa)}$ modular exponentiations. If tags of length $\tilde{O}(\log\kappa)$ are provided, the overhead is only $r+{\tilde{O}(\log^2\kappa)}$ modular exponentiations. The only previous concurrent non-malleable zero-knowledge (under non-adaptive inputs) was achieved by Barak, Prabhakaran and Sahai [4]. Their construction, however, mainly focuses on a feasibility result rather than efficiency, and requires expensive ${\mathcal{NP}}$-reductions.