Bisection and squares in genus 2

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• 作者：Josep M. Miret^a ; ^{miret@matematica.udl.cat" class="auth_mail" title="E-mail the corresponding author} ; Jordi Pujolà ; s^a ; ^{jpujolas@matematica.udl.cat" class="auth_mail" title="E-mail the corresponding author} ; Nicolas Thé ; riault^b ; ^{ntheriau@ubiobio.cl" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11G20 ; 11T71 ; 14G50 ; 14H40 ; 14H45
• 刊名：Finite Fields and Their Applications
• 出版年：2015
• 出版时间：November 2015
• 年：2015
• 卷：36
• 期：Complete
• 页码：170-188
• 全文大小：426 K

文摘

We show how to compute the pre-images of multiplication-by-2 in Jacobians of genus 2 curves C:y²=f(x)$\mathrm{C}:y^2=f(x)$ over F_q${\mathbb{F}}_q$ with q   odd. We characterize D=[u(x),v(x)]∈2Jac(C)(F_q)$D=[u(x),v(x)]\in 2\mathrm{Jac}(\mathrm{C})({\mathbb{F}}_q)$ in terms of the quadratic character of u(x)$u(x)$ at the roots of f(x)$f(x)$ in imaginary models, and in terms of the quadratic character of the quotients of u(x)$u(x)$ at pairs of roots of f(x)$f(x)$ in real models. Our method reduces the problem to the computation of at most 5 square roots over the splitting field of f(x)$f(x)$ plus the solution of a system of linear equations.