### A New Twofold Cornacchia-Type Algorithm and Its Applications

Bei Wang, Yi Ouyang, Honggang Hu, and 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.

Available format(s)
Category
Public-key cryptography
Publication info
Preprint. MINOR revision.
Keywords
Elliptic curvesHyperelliptic curvesEndomorphisms4-GLV decompositionsTwofold Cornacchia-type algorithms.
Contact author(s)
wangbei @ mail ustc edu cn
History
2021-05-12: last of 2 revisions
See all versions
Short URL
https://ia.cr/2021/090

CC BY

BibTeX

@misc{cryptoeprint:2021/090,
author = {Bei Wang and Yi Ouyang and Honggang Hu and Songsong Li},
title = {A New Twofold Cornacchia-Type Algorithm and Its Applications},
howpublished = {Cryptology ePrint Archive, Paper 2021/090},
year = {2021},
note = {\url{https://eprint.iacr.org/2021/090}},
url = {https://eprint.iacr.org/2021/090}
}
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