We study the following two related questions: - What are the minimal computational resources required for general secure multiparty computation in the presence of an honest majority? - What are the minimal resources required for two-party primitives such as zero-knowledge proofs and general secure two-party computation? We obtain a nearly tight answer to the first question by presenting a perfectly secure protocol which allows n players to evaluate an arithmetic circuit of size s by performing a total of O(slogs log~2 n) arithmetic operations, plus an additive term which depends (polynomially) on n and the circuit depth, but only logarithmically on s. Thus, for typical large- scale computations whose circuit width is much bigger than their depth and the number of players, the amortized overhead is just polylogarith- mic in n and s. The protocol provides perfect security with guaranteed output delivery in the presence of an active, adaptive adversary cor rupting a (1/3 - ε) fraction of the players, for an arbitrary constant ε > 0 and sufficiently large n. The best previous protocols in this setting could only offer computational security with a computational overhead of poly(k, log n, log s), where k is a computational security parameter, or perfect security with a computational overhead of O(n log n). We then apply the above result towards making progress on the second question. Concretely, under standard cryptographic assumptions, we obtain zero-knowledge proofs for circuit satisfiability with 2~(-k) soundness error in which the amortized computational overhead per gate is only poly logarithmic in k, improving over the ω(k) overhead of the best previous protocols. Under stronger cryptographic assumptions, we obtain similar results for general secure two-party computation.
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