Isogenies and the Discrete Logarithm Problem in Jacobians of Genus 3 Hyperelliptic Curves,
Isogenies and the Discrete Logarithm Problem in Jacobians of Genus 3 Hyperelliptic Curves,
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DOI：
10.1007/s00145-009-9038-1
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发表时间：
2008-04
影响因子：
3
通讯作者：
Benjamin A. Smith
中科院分区：
文献类型：
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作者：
Benjamin A. Smith
我们描述了使用显式的同源性来翻译的实例的离散对数问题（DLP）的超椭圆属3曲线的雅可比矩阵的非超椭圆属3曲线，在那里他们是容易受到更快的指数演算攻击。我们提供了显式公式的同构与核同构的（<$/2 <$）3（在代数封闭的基础领域）的任何超椭圆亏格3曲线在一个领域的特征不2或3。对于特征p>3的有限域上所有亏格为3的超椭圆曲线的正分数，这些同构是有理的。在合理的假设下，我们的构造给出了一个明确和有效的减少的DLP的实例从超椭圆到非超椭圆雅可比约18.57%的超椭圆属3曲线在一个给定的有限域。最后，我们讨论这些想法扩展到更一般的内核的同源。
We describe the use of explicit isogenies to translate instances of the Discrete Logarithm Problem (DLP) from Jacobians of hyperelliptic genus 3 curves to Jacobians of non-hyperelliptic genus 3 curves, where they are vulnerable to faster index calculus attacks. We provide explicit formulae for isogenies with kernel isomorphic to (ℤ/2ℤ)3(over an algebraic closure of the base field) for any hyperelliptic genus 3 curve over a field of characteristic not 2 or 3. These isogenies are rational for a positive fraction of all hyperelliptic genus 3 curves defined over a finite field of characteristicp>3. Subject to reasonable assumptions, our constructions give an explicit and efficient reduction of instances of the DLP from hyperelliptic to non-hyperelliptic Jacobians for around 18.57% of all hyperelliptic genus 3 curves over a given finite field. We conclude with a discussion on extending these ideas to isogenies with more general kernels.