Complete group law for genus 2 Jacobians on Jacobian coordinates

Abstract

This manuscript provides complete inversion-free and explicit group law formulas in Jacobian coordinates for the genus 2 hyperelliptic curves of the form y 2 = x 5 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y<^>2 = x<^>5+a_3 x<^>3+a_2 x<^>2+a_1 x+a_0$$\end{document} over a field K with char ( K ) not equal 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{char}(K) \ne 2$$\end{document} . The formulas do not require the use of polynomial arithmetic operations such as resultant mod or gcd computations but only operations in K.