Computing gluing and splitting $$(\ell ,\ell )$$ -isogenies

IF 1.4 2区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Designs, Codes and Cryptography Pub Date : 2024-05-02 DOI:10.1007/s10623-024-01413-x

Song Tian

引用次数: 0

Abstract

We give algorithms to compute $(\ell ,\ell )$-isogenies between Jacobians of genus 2 curves and products of elliptic curves in time of ${\tilde{O}}(\ell ^2)$ basic field operations, where $\ell $ is an odd prime different from the characteristic of the field. The method relies on the notion of a normal Weil set of Weil functions due to Shepherd-Barron, and works for the case of computing $(\ell ,\ell )$-isogenies between Jacobians of genus 2 curves.

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计算粘合与分裂 $$(\ell ,\ell )$$ - 等因关系

我们给出了用${\tilde{O}}(\ell ^2)$基本场运算的时间计算属2曲线的雅各布数和椭圆曲线乘积之间的$(\ell ,\ell)$-异或的算法，其中$\ell $是不同于场特征的奇素数。该方法依赖于谢泼德-巴伦（Shepherd-Barron）提出的韦尔函数的正常韦尔集的概念，并且适用于计算属2曲线的雅各比之间的$(\ell ,\ell)$-同源性的情况。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Designs, Codes and Cryptography 工程技术-计算机：理论方法

CiteScore

2.80

自引率

12.50%

发文量

157

审稿时长

16.5 months

期刊介绍： Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines. The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome. The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas. Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.