Paper 2021/220

A New Twofold Cornacchia-Type Algorithm

Bei Wang, Yi Ouyang, Songsong Li, and Honggang Hu


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 curve. Besides, we give some applications of our new algotithm in some cuvres not considered in Longa and Sica's algorithm.

Available format(s)
Public-key cryptography
Publication info
Preprint. MINOR revision.
Elliptic curves4-GLV decompositionsTwofold Cornacchia-type algorithm
Contact author(s)
wangbei @ mail ustc edu cn
2021-03-04: last of 3 revisions
2021-03-02: received
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Short URL
Creative Commons Attribution


      author = {Bei Wang and Yi Ouyang and Songsong Li and Honggang Hu},
      title = {A New Twofold Cornacchia-Type Algorithm},
      howpublished = {Cryptology ePrint Archive, Paper 2021/220},
      year = {2021},
      note = {\url{}},
      url = {}
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