Paper 2021/090
A New Twofold CornacchiaType 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 4GLV decompositions. The algorithm consists of two subalgorithms, 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 subalgorithm can be replaced by another Euclidean domain $\mathbb{Z}[\omega]$ $(\omega=\frac{1+\sqrt{3}}{2})$. As a consequence, we design a new twofold Cornacchiatype 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.
Metadata
 Available format(s)
 Category
 Publickey cryptography
 Publication info
 Preprint. MINOR revision.
 Keywords
 Elliptic curvesHyperelliptic curvesEndomorphisms4GLV decompositionsTwofold Cornacchiatype algorithms.
 Contact author(s)
 wangbei @ mail ustc edu cn
 History
 20210512: last of 2 revisions
 20210127: received
 See all versions
 Short URL
 https://ia.cr/2021/090
 License

CC BY
BibTeX
@misc{cryptoeprint:2021/090, author = {Bei Wang and Yi Ouyang and Honggang Hu and Songsong Li}, title = {A New Twofold CornacchiaType 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} }