Characterizing Bent Concatenations Outside the Completed Maiorana-McFarland Class

核心概念

Bent functions can be constructed as concatenations of other bent, semi-bent, or five-valued spectra functions. The key question is when such bent concatenations do not belong to the completed Maiorana-McFarland class. This article provides a full characterization of the necessary and sufficient conditions for bent concatenations to be outside the Maiorana-McFarland class.

摘要

The article investigates the properties of bent concatenations and their relationship to the completed Maiorana-McFarland class of bent functions.
Key highlights:

Bent functions can be written as concatenations of two complementary semi-bent functions (f = f1||f2) or four bent, semi-bent, or five-valued spectra functions (f = f1||f2||f3||f4).
The authors provide a full characterization of the structure of M-subspaces for these concatenations, which allows them to specify the necessary and sufficient conditions for the bent concatenation to be outside the Maiorana-McFarland class.
For the concatenation f = f1||f2, the bent function f is outside the Maiorana-McFarland class if the functions f1 and f2 do not share a common (k+1)-dimensional M-subspace, and for every k-dimensional M-subspace V of both f1 and f2, there exists a vector a in V such that Daf1(z) + Daf2(z+u) is non-zero for some z.
For the concatenation f = f1||f2||f3||f4, the bent function f is outside the Maiorana-McFarland class if the functions f1, f2, f3, f4 do not share a common (n/2+1)-dimensional M-subspace, and there are no common (n/2)-dimensional M-subspaces V of f1, f2, f3, f4 satisfying certain conditions.
The authors also propose several explicit design methods for constructing bent functions outside the Maiorana-McFarland class, particularly in the case of f = g||h||g||(h+1), where g and h are bent functions.

統計資料

None.

引述

None.

從以下內容提煉的關鍵洞見

When does a bent concatenation not belong to the completed Maiorana-McFarland class?

by Sadmir Kudin... 於 arxiv.org 04-26-2024

https://arxiv.org/pdf/2404.16220.pdf

深入探究

How can the insights from this work be extended to construct bent functions outside the Maiorana-McFarland class for a wider range of concatenation structures beyond the ones considered in this article?

The insights gained from the characterization of M-subspaces for bent functions can be extended to construct bent functions outside the Maiorana-McFarland class for various concatenation structures. One approach is to explore different combinations of semi-bent, bent, or five-valued spectra functions in the concatenation process. By varying the types of functions used in the concatenation and analyzing their M-subspace properties, it is possible to identify patterns and conditions that lead to bent functions outside the M# class.
Additionally, researchers can investigate more complex concatenation structures involving multiple functions and study the interplay between their M-subspaces. By systematically analyzing the M-subspace properties of these concatenated functions, new criteria and design principles can be established to guide the construction of bent functions outside the Maiorana-McFarland class. This extension of the work can lead to a broader understanding of the structural characteristics that define bent functions and facilitate the creation of diverse cryptographic primitives with enhanced properties.

What are the implications of the characterization of M-subspaces for the design of cryptographic primitives based on bent functions, such as in the context of block ciphers or stream ciphers?

The characterization of M-subspaces plays a crucial role in the design of cryptographic primitives based on bent functions, particularly in the development of block ciphers and stream ciphers. Understanding the M-subspace properties of bent functions allows cryptographers to assess their algebraic structures, non-linearity, and resistance to cryptanalysis techniques.
In the context of block ciphers, the knowledge of M-subspaces can guide the selection and combination of bent functions to achieve desirable cryptographic properties such as confusion and diffusion. By leveraging the insights from M-subspace analysis, designers can create substitution boxes (S-boxes) with enhanced cryptographic strength and resistance against differential and linear cryptanalysis.
For stream ciphers, the characterization of M-subspaces can inform the design of non-linear feedback functions that exhibit desirable properties for generating secure and unpredictable keystreams. By incorporating bent functions with specific M-subspace characteristics, stream ciphers can achieve high levels of security and randomness, essential for protecting sensitive data in communication systems.
Overall, the insights from M-subspace characterization provide a foundation for designing cryptographic primitives that offer robust security guarantees and withstand various cryptanalytic attacks in both block ciphers and stream ciphers.

Are there any connections between the properties of M-subspaces and the computational complexity of problems related to bent functions, such as the problem of determining the algebraic degree of a bent function?

The properties of M-subspaces are closely linked to the computational complexity of problems related to bent functions, including determining the algebraic degree of a bent function. The existence and structure of M-subspaces impact the algebraic properties and non-linearity of bent functions, which in turn influence the complexity of algebraic computations and analyses involving these functions.
In the context of determining the algebraic degree of a bent function, the presence of M-subspaces can provide valuable insights into the structure of the function and its algebraic normal form. M-subspaces affect the distribution of Walsh coefficients, the existence of linear structures, and the overall algebraic complexity of the function, all of which are crucial factors in determining its algebraic degree.
Furthermore, the characterization of M-subspaces can aid in identifying specific patterns or properties that simplify or complicate the computation of the algebraic degree of bent functions. By analyzing the M-subspace structure, researchers can develop efficient algorithms and techniques for algebraic degree computation, leveraging the inherent properties of M-subspaces to streamline the process and optimize computational complexity.
Overall, the properties of M-subspaces play a significant role in shaping the computational complexity of problems related to bent functions, offering valuable insights into the algebraic structure and behavior of these cryptographic primitives.