On Boolean functions derived from linear maps over $$\mathbb {Z}_4$$ and their application to secret sharing

IF 1.4 2区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Designs, Codes and Cryptography Pub Date : 2024-08-16 DOI:10.1007/s10623-024-01478-8

Deepak Agrawal, Srinivasan Krishnaswamy, Smarajit Das

引用次数: 0

Abstract

The Gray map converts a symbol in $\mathbb {Z}_4$ to a pair of binary symbols. Therefore, under the Gray map, a linear function from $\mathbb {Z}_4^n$ to $\mathbb {Z}_4$ gives rise to a pair of boolean functions from $\mathbb {F}_2^{2n}$ to $\mathbb {F}_2$. This paper studies such boolean functions. We state and prove a condition for the nonlinearity of such functions and derive closed-form expressions for them. Further, results related to the mutual information between random variables that satisfy such expressions have been derived. These results are then used to construct a couple of nonlinear boolean secret sharing schemes. These schemes are then analyzed for their closeness to ‘perfectness’ and their ability to resist ‘Tompa–Woll’-like attacks.

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论从 $$\mathbb {Z}_4$$ 上的线性映射派生的布尔函数及其在秘密共享中的应用

格雷映射将 $\mathbb {Z}_4$ 中的一个符号转换成一对二进制符号。因此，在格雷映射下，从$\mathbb {Z}_4^n$ 到$\mathbb {Z}_4$的线性函数会产生一对从$\mathbb {F}_2^{2n}$ 到$\mathbb {F}_2$的布尔函数。本文研究的就是这样的布尔函数。我们指出并证明了这类函数的非线性条件，并推导出了它们的闭式表达式。此外，我们还推导出了与满足此类表达式的随机变量之间的互信息相关的结果。然后，我们利用这些结果构建了几个非线性布尔秘密共享方案。然后分析了这些方案与 "完美性 "的接近程度以及抵御类似 "Tompa-Woll "攻击的能力。

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.

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