对于一般化的布尔函数的构造和等价性.
Ayça Çeşmelioğlu1, Wilfried Meidl2
1Özyeğin University, Nişantepe Mah. Orman Sk., 34794 Çekmeköy, Istanbul, Turkey.
Cryptography and communications : discrete structures, Boolean functions and sequences
|October 23, 2025
概括
这项研究扩展了对有限场的向量空间中的函数等价值 (EA和CCZ) 的理解,为一般化的曲函数引入了新的构造. 该研究提供了生成这些关键加密组件的大集方法.
科学领域:
- 数学理论 数学理论
- 抽象代数 抽象代数
- 密码学 密码学 密码学
背景情况:
- 对函数等值的研究,特别是EA等值和CCZ等值,对于理解布尔函数的属性及其在密码学中的应用至关重要.
- 之前的工作启动了对这些等价函数的研究,这些函数在有限场 F_p 上映映射向量空间到周期组 Z_{p^k},对 p=2 和 k=2.2 进行了具体的发现.
研究的目的:
- 扩展EA等效和CCZ等效的结果到从V_n^{(p)}到Z_{p^k}的更广泛的函数类.
- 开发通用曲函数的构造,特别是奇数 p 和偶数 n 的 p=2 的构造.
- 介绍一种方法,用于构建从具有较小域的函数中对p=2和奇数n的概括曲面函数.
主要方法:
- 扩展现有的等价性研究到更大的函数类别.
- 直接和半直接的总和结构的应用.
- 使用二次曲函数构造方法.
- 基于具有缩小尺寸的函数,开发一种针对一般化的曲函数的新构造.
主要成果:
- 该研究成功地将EA等价和CCZ等价的分析扩展到更一般的函数类.
- 介绍了构建通用曲函数的新方法,创建了这些函数的大 affine 空间.
- 为从V_n^{(2)}到Z_{2^k} (n奇数) 的概括曲面函数提供了一个特定的构造,使用从V_{n-1}^{(2)}到Z_{2^{k-1}}的概括曲面函数.
结论:
- 这项研究加深了对有限场中的函数等值及其与曲函数的关系的理解.
- 提出的构造提供了宝贵的工具,用于生成广义曲函数的家族,这对于密码设计很重要.
- 这些发现有助于曲函数理论及其在编码理论和密码学中的实际构造.
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