A theoretical and empirical evaluation of an algorithm for self-healing computation

In the problem of reliable multiparty computation (RMC), there are n parties, each with an individual input, and the parties want to jointly compute a function f over n inputs; note that it is not required to keep the inputs private. The problem is complicated by the fact that an omniscient adversary controls a hidden fraction of the parties. We describe a self-healing algorithm for this problem. In particular, for a fixed function f, with n parties and m gates, we describe how to perform RMC repeatedly as the inputs to f change. Our algorithm maintains the following properties, even when an adversary controls up to t≤(14-ϵ)n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \le (\frac{1}{4} - \epsilon ) n$$\end{document} parties, for any constant ϵ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon >0$$\end{document}. First, our algorithm performs each reliable computation with the following amortized resource costs: O(m+nlogn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(m + n \log n)$$\end{document} messages, O(m+nlogn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(m + n \log n)$$\end{document} computational operations, and O(ℓ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\ell )$$\end{document} latency, where ℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document} is the depth of the circuit that computes f. Second, the expected total number of corruptions is O(t(log∗m)2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(t (\log ^*{m})^2)$$\end{document}, after which the adversarially controlled parties are effectively quarantined so that they cause no more corruptions. Empirical results show that our algorithm can reduce message cost by a factor of 432 when compared with algorithms that are not self-healing.