$(\ell,\ell)$-isogeny from a product of ellptic curves

このパッケージはSCIS2025で吉住(九州大学)が発表したアルゴリズムのSageMathによる実装です. アルゴリズムの詳しい内容は(参加者のみが見れる)予稿を参照します.

This package contains the implementation of the algorithm in SageMath presented at SCIS2025 by Ryo Yoshizumi. The detailed contents of the algorithm can be found in the proceedings, which are accessible only to participants.

Content

In this package, there is functions to compute $(\ell,\ell)$-isogeny from a product of ellptic curves. The main module is func_isogeny.py. For the overall flow, please refer to test.py.

This code is written in SageMath.

One Example

By writing the following command, you can compute one example of $(\ell,\ell)$-isogeny from $E_0\times E_0$.

% sage example.py

Here, the base field is $\mathbb{F}_{p^2}$ where $p=276154505650672190920223$, $\ell=11$, and $E_0 : y^2=x^3+x$.

The kernel $K\subset (E_0\times E_0)[\ell]$ is randomly generalted, and we write a basis of $K$ by $e_1=(e_1^{(1)},e_1^{(2)}),e_2=(e_2^{(1)},e_2^{(2)})$.

Evaluating point is $x=(x^{(1)},0_{E_0})$ where $x^{(1)}\in E_0$ is a random point.

The output is as follows:

p: 276154505650672190920223

ell: 11

E_0:
Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 276154505650672190920223^2

e1=(e1^(1),e1^(2)):
((210831735326664260671440*z2 + 87198234199232688535215 : 133255005542521365746194*z2 + 164977575746837862238202 : 1), (83375910444643963132697*z2 + 200948769993679858502156 : 144292860645864077727680*z2 + 148419487573216577660120 : 1))

e2=(e2^(1),e2^(2)):
((98830049968673606594523 : 242317345512027435674284*z2 + 49548615497841843339319 : 1), (177324455681998584325700 : 216601895301696610344217 : 1))

x=(x^(1),0):
((106189123109636747922251*z2 + 248968408709573238167579 : 151118498536791259433057*z2 + 87307954336256715411728 : 1), (0 : 1 : 0))

theta-null point of the codomain:
[112027156618648925977355*z2 + 147438371256409604632158, 69068842046189523671338*z2 + 222257352160280440335182, 275376810320619078414513*z2 + 183273802569472479972437, 86875977424383747210632*z2 + 210469468339113327744966, 1]

theta coordinate of the image f(x):
[51118548117130760667387*z2 + 22524482140569545457684, 179878823554441321130688*z2 + 205994074250712649272564, 198607624210760764176903*z2 + 126880054694919192506851, 103634261912850587289929*z2 + 224536590004812764733705, 1]

Author

• Name: Ryo Yoshizumi
• Affiliation: Kyushu University
• Email: [email protected]