Antiderivative functions over \(\mathbb {F}_{2^n}\)

详细信息    查看全文

• 作者：Valentin Suder
• 关键词：Derivative functions ; Higher order derivative functions ; Antidifferentiation over \(\mathbb {F}_{2^n}\) ; Antiderivative functions ; Linear structure ; Quadratic APN functions
• 刊名：Designs, Codes and Cryptography
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：82
• 期：1-2
• 页码：435-447
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Combinatorics; Coding and Information Theory; Data Structures, Cryptology and Information Theory; Data Encryption; Discrete Mathematics in Computer Science; Information and Communication, Circuits;
• 出版者：Springer US
• ISSN：1573-7586
• 卷排序：82

文摘

In this paper, we use a linear algebra point of view to describe the derivatives and higher order derivatives over \(\mathbb {F}_{2^n}\). On one hand, this new approach enables us to prove several properties of these functions, as well as the functions that have these derivatives. On the other hand, we provide a method to construct all of the higher order derivatives in given directions. We also demonstrate some properties of the higher order derivatives and their decomposition as a sum of functions with 0-linear structure. Moreover, we introduce a criterion and an algorithm to realize discrete antidifferentiation of vectorial Boolean functions. This leads us to define a new equivalence of functions, that we call differential equivalence, which links functions that share the same derivatives in directions given by some subspace. Finally, we discuss the importance of finding 2-to-1 functions.