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地图的代数属性

Jan Schoone1, Joan Daemen1

  • 1Digital Security, Radboud University, Nijmegen, The Netherlands.

Designs, codes, and cryptography
|July 29, 2024
PubMed
概括
此摘要是机器生成的。

本研究分析了密码学中使用的布尔图的代数属性. 研究人员发现了它.

关键词:
布尔式地图是一个布尔式地图.千千千千千千千千千千千千千千千千千千千千千千千千千千千千千千密码学 密码学 密码学 密码学多项式地图的多项式地图对称密码学对称密码学

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科学领域:

  • 密码学 密码学 密码学 密码学
  • 抽象代数 抽象代数
  • 数学理论 数学理论

背景情况:

  • 布尔图 $x \mapsto x^{2^n-2}$ 是几个现代加密代码的核心组件.
  • 这些排列包括Keccak-f (SHA-3),ASCON (NIST轻量级比赛获胜者),Xoodoo,Rasta和地下.
  • 了解该地图的代数属性对于加密分析和安全加密系统的设计至关重要.

研究的目的:

  • 在有限字段上研究布尔图 $x \mapsto x^{2^n-2}$ 的代数特征.
  • 确定该图表表现为功率函数的条件,并分析其多项式表示.
  • 检查反向图的属性及其对字段扩展的行为.

主要方法:

  • 将布尔图表表示为有限字段上的单变多项式.
  • 分析地图为功率函数的条件,使用矢量同态.
  • 计算局限于多项式的稀疏性,度和单变量表示数.
  • 在反向图的多项式表示中计算单项数的数量.
  • 在 $\mathbb{F}_{2^n}$ 的字段扩展上研究地图作为多项式地图的行为.

主要成果:

  • 布尔图 $x \mapsto x^{2^n-2}$ 是一个功率函数,如果,只有如果 $n=1$.
  • 计算了单变量表示的稀疏性,程度和数量的边界.
  • 在反向图中,给定度的单项数的数量与二项系数一致.
  • 图 $x \mapsto x^{2^n-2}$ 在扩展度可被 2 或 3 分的 $\mathbb{F}_{2^n}$ 字段扩展度上不会产生一个 bijection.

结论:

  • 该研究提供了现代对称密钥密码学中一个关键组件的综合代数分析.
  • 这些发现为布尔图的结构和局限性及其反面提供了洞察力.
  • 提出了一个猜想,即规则 $x \mapsto x^{2^n-2}$ 不定义在任何 $\mathbb{F}_{2^n}$ 的扩展字段上的 bijection.