**Perfectly Secure Multiparty Computation and the Computational Overhead of Cryptography**

*Ivan Damgård and Yuval Ishai and Mikkel Krøigaard*

**Abstract: **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 $\O(s\log s\log^2 n)$ 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 polylogarithmic in $n$ and $s$. The protocol provides perfect security with guaranteed output delivery in the presence of an active, adaptive adversary corrupting a $(1/3-\epsilon)$ fraction of the players, for an arbitrary constant $\epsilon>0$ 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 {\em polylogarithmic} in $k$, improving over the $\omega(k)$ overhead of the best previous protocols. Under stronger cryptographic assumptions, we obtain similar results for general secure two-party computation.

**Category / Keywords: **cryptographic protocols / multiparty computation

**Publication Info: **Full version of Eurocrypt 2010 paper

**Date: **received 26 Feb 2010, last revised 26 Feb 2010

**Contact author: **mk at cs au dk

**Available format(s): **PDF | BibTeX Citation

**Version: **20100301:232359 (All versions of this report)

**Short URL: **ia.cr/2010/106

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