Constant-Depth Circuits for Arithmetic in Finite Fields of Characteristic Two

We study the complexity of arithmetic in finite fields of characteristic two, \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}. We concentrate on the following two problems: – Iterated Multiplication: Given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha_1,...,\alpha_t \in \mathbb{F}_{2^n}$\end{document}, compute α1 ⋯ αt. – Exponentiation: Given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha \in \mathbb{F}_{2^n}$\end{document} and a t-bit integer k, compute αk.