In　MILCOM　2015,　Kelly　et　al.　proposed　the　authentication　encryption　algorithm　MK-3,　which　applied　the　16-bit　S-box.　This　paper　aims　to　implement　the　16-bit　S-box　with　less　circuit　area.　First,　we　classified　the　irreducible　polynomials　over　F2n\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\mathbb　{F}_{2＜^＞n}$$\end{document}　into　three　kinds.　Then　we　compared　the　logic　gates　required　for　multiplication　over　the　finite　field　constructed　by　the　three　types　of　irreducible　polynomials.　According　to　the　comparison　result,　we　constructed　the　composite　fields,　F(24)2\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\mathbb　{F}_{(2＜^＞4)＜^＞2}$$\end{document}　and　F(28)2\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\mathbb　{F}_{(2＜^＞8)＜^＞2}$$\end{document}.　Based　on　the　isomorphism　of　finite　fields,　the　operations　over　F216\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\mathbb　{F}_{2＜^＞{16}}$$\end{document}　can　be　conducted　over　F(28)2\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\mathbb　{F}_{(2＜^＞8)＜^＞2}$$\end{document}.　Similarly,　elements　over　F28\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\mathbb　{F}_{2＜^＞8}$$\end{document}　can　be　mapped　to　the　corresponding　elements　over　F(24)2\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\mathbb　{F}_{(2＜^＞4)＜^＞2}$$\end{document}.　Next,　the　SAT　solver　was　used　to　optimize　the　operations　over　smaller　field　F24\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\mathbb　{F}_{2＜^＞4}$$\end{document}.　At　last,　the　architecture　of　the　optimized　MK-3　S-box　was　worked　out.　Compared　with　the　implementation　proposed　by　the　original　designer,　the　circuit　area　of　the　MK-3　S-box　in　this　paper　is　reduced　by　at　least　55.9%.