Use of insoluble polymer supports in organic synthesis. 9. Synthesis of unsymmetrical carotenoids on solid phases

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? 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 (s log s log2 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 polylogarithmic in n and s. The protocol provides perfect security with guaranteed output delivery in the presence of an active, adaptive adversary corrupting a (1/3 ¿ ") fraction of the players, for an arbitrary constant ¿ 0 and succiently 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 polylogarithmic in k, improving over the ¿ (k) overhead of the best previous protocols. Under stronger cryptographic assumptions, we obtain similar results for general secure two-party computation.