SIAM Journal on Computing, Volume 50, Issue 1, Page 68-97, January 2021.
We show that any multiparty functionality can be evaluated using a 2-round protocol with perfect correctness and perfect semihonest security, provided that the majority of parties are honest. This settles the round complexity of information-theoretic semihonest multiparty computation, resolving a longstanding open question [Y. Ishai and E. Kushilevitz, Randomizing polynomials: A new representation with applications to round-efficient secure computation, in Proceedings of the 41st Annual Symposium on Foundations of Computer Science FOCS 2000, IEEE Computer Society, 2000, pp. 294--304]. The protocol is efficient for ${NC}^1$ functionalities. Furthermore, given black-box access to a one-way function, the protocol can be made efficient for any polynomial functionality, at the cost of only guaranteeing computational security. Our results are based on a new notion of multiparty randomized encoding which extends and relaxes the standard notion of randomized encoding of functions [Y. Ishai and E. Kushilevitz, Randomizing polynomials: A new representation with applications to round-efficient secure computation, in Proceedings of the 41st Annual Symposium on Foundations of Computer Science FOCS 2000, IEEE Computer Society, 2000, pp. 294--304]. The property of a multiparty randomized encoding (MPRE) is that if the functionality $g$ is an encoding of the functionality $f$, then for any (permitted) coalition of players, their respective outputs and inputs in $g$ allow them to simulate their respective inputs and outputs in $f$, without learning anything else, including the other outputs of $f$. We further introduce a new notion of effective degree, and show that the round complexity of a functionality $f$ is characterized by the degree of its MPRE. We construct degree-2 MPREs for general functionalities in several settings under different assumptions, and use these constructions to obtain 2-round protocols. Our constructions also give rise to new protocols in the client-server model with optimal round complexity.

中文翻译：

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两轮完美的安全计算

SIAM计算杂志，第50卷，第1期，第68-97页，2021年1月。
我们证明，只要大多数参与者都是诚实的，就可以使用具有正确性和完美半诚实性的2轮协议评估任何多方功能。这就解决了信息理论上半诚实的多方计算的复杂性，解决了一个长期存在的开放性问题。Ishai和E. Kushilevitz，“多项式的随机化：一种适用于舍入有效安全计算的新表示形式”，在第41届计算机科学基础年度研讨会论文集FOCS 2000中，IEEE计算机协会，2000年，第294--304页]。该协议对于$ {NC} ^ 1 $功能非常有效。此外，给定黑匣子访问单向函数的权限，该协议对于任何多项式功能都可以变得高效，而仅以保证计算安全性为代价。我们的结果基于多方随机编码的新概念，该概念扩展和放松了函数随机编码的标准概念[Y. Ishai和E. Kushilevitz，“多项式的随机化：一种适用于舍入有效安全计算的新表示形式”，在第41届计算机科学基础年度研讨会论文集FOCS 2000中，IEEE计算机协会，2000年，第294--304页]。多方随机编码（MPRE）的特性是，如果功能$ g $是功能$ f $的编码，则对于任何（允许的）玩家联盟，其各自的输出和$ g $的输入都允许他们在$ f $中模拟它们各自的输入和输出，而无需学习其他任何东西，包括$ f $的其他输出。我们进一步引入有效程度的新概念，并显示了功能$ f $的总体复杂度由其MPRE的程度来表征。我们在不同的假设下，在几种情况下为通用功能构造了2级MPRE，并使用这些构造来获得2轮协议。我们的构造还产生了具有最佳回合复杂度的客户端-服务器模型中的新协议。