Secure multi-party computation with legally-enforceable fairness

Fairness is a security notion of secure computation and cannot always be achieved if an adversary corrupts a majority of parties in standard settings. Lindell (CT-RSA 2008) showed that imposing a monetary penalty on an adversary can circumvent the impossibility. He formalized such a security notion as "legally enforceable fairness" for the two-party setting based on the ideal trusted bank functionality and showed a protocol achieving the requirements. Based on the same framework, we introduce secure multi-party computation with legally enforceable fairness that is applicable for an arbitrary number of parties. Further, we propose two protocols that realize our introduced functionality. The first one achieves O(n) rounds and O(n alpha)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n \alpha )$$\end{document} fees, where n is the number of parties, and alpha\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} is a parameter for the penalty amount. The fee refers to the balance amount in the bank required at the beginning of the protocol, which evaluates the difficulty of participating in the protocol in a financial sense. The second one achieves O(1) rounds and O(n2 alpha)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n<^>2 \alpha )$$\end{document} fees.