Paper 2021/090

A New Twofold Cornacchia-Type Algorithm and Its Applications

Bei Wang; Yi Ouyang; Honggang Hu ; Songsong Li

Abstract

We focus on exploring more potential of Longa and Sica's algorithm (ASIACRYPT 2012), which is an elaborate iterated Cornacchia algorithm that can compute short bases for 4-GLV decompositions. The algorithm consists of two sub-algorithms, the first one in the ring of integers $\mathbb{Z}$ and the second one in the Gaussian integer ring $\mathbb{Z}[i]$. We observe that $\mathbb{Z}[i]$ in the second sub-algorithm can be replaced by another Euclidean domain $\mathbb{Z}[\omega]$ $(\omega=\frac{-1+\sqrt{-3}}{2})$. As a consequence, we design a new twofold Cornacchia-type algorithm with a theoretic upper bound of output $C\cdot n^{1/4}$, where $C=\frac{3+\sqrt{3}}{2}\sqrt{1+|r|+|s|}$ with small values $r, s$ given by the curves. The new twofold algorithm can be used to compute $4$-GLV decompositions on two classes of curves. First it gives a new and unified method to compute all $4$-GLV decompositions on $j$-invariant $0$ elliptic curves over $\mathbb{F}_{p^2}$. Second it can be used to compute the $4$-GLV decomposition on the Jacobian of the hyperelliptic curve defined as $\mathcal{C}/\mathbb{F}_{p}:y^{2}=x^{6}+ax^{3}+b$, which has an endomorphism $\phi$ with the characteristic equation $\phi^2+\phi+1=0$ (hence $\mathbb{Z}[\phi]=\mathbb{Z}[\omega]$). As far as we know, none of the previous algorithms can be used to compute the $4$-GLV decomposition on the latter class of curves.