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

Sadmir Kudin, Enes Pasalic, Alexandr Polujan, Fengrong Zhang

TL;DR

This work provides a complete characterization of when bent Boolean functions formed by concatenating smaller components lie outside the completed Maiorana–McFarland class $\mathcal{M}^\#$. By deriving precise structures of $\mathcal{M}$-subspaces for two- and four-function concatenations, the authors establish necessary and sufficient conditions for outside-$\mathcal{M}^\#$, notably tying the property to the nonexistence of common $\mathcal{M}$-subspaces among the constituent functions. They apply these results to the special form $f=g||h||g||(h+1)$ to show that outside-$\mathcal{M}^\#$ design is achievable whenever $g$ and $h$ lack a shared $(n/2)$-dimensional $\mathcal{M}$-subspace, and provide Korsakova-type constructions and affine-transform-based methods to generate such bent functions for various $n$, including $n\ge 10$. The findings offer practical design strategies for producing bent functions with cryptographic relevance beyond $\mathcal{M}^\#$, including plentiful 8-variable and higher-dimensional examples.

Abstract

Every Boolean bent function $f$ can be written either as a concatenation $f=f_1||f_2$ of two complementary semi-bent functions $f_1,f_2$; or as a concatenation $f=f_1||f_2||f_3||f_4$ of four Boolean functions $f_1,f_2,f_3,f_4$, all of which are simultaneously bent, semi-bent, or 5-valued spectra-functions. In this context, it is essential to ask: When does a bent concatenation $f$ (not) belong to the completed Maiorana-McFarland class $\mathcal{M}^\#$? In this article, we answer this question completely by providing a full characterization of the structure of $\mathcal{M}$-subspaces for the concatenation of the form $f=f_1||f_2$ and $f=f_1||f_2||f_3||f_4$, which allows us to specify the necessary and sufficient conditions so that $f$ is outside $\mathcal{M}^\#$. Based on these conditions, we propose several explicit design methods of specifying bent functions outside $\mathcal{M}^\#$ in the special case when $f=g||h||g||(h+1)$, where $g$ and $h$ are bent functions.

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

TL;DR

This work provides a complete characterization of when bent Boolean functions formed by concatenating smaller components lie outside the completed Maiorana–McFarland class . By deriving precise structures of -subspaces for two- and four-function concatenations, the authors establish necessary and sufficient conditions for outside-, notably tying the property to the nonexistence of common -subspaces among the constituent functions. They apply these results to the special form to show that outside- design is achievable whenever and lack a shared -dimensional -subspace, and provide Korsakova-type constructions and affine-transform-based methods to generate such bent functions for various , including . The findings offer practical design strategies for producing bent functions with cryptographic relevance beyond , including plentiful 8-variable and higher-dimensional examples.

Abstract

Every Boolean bent function can be written either as a concatenation of two complementary semi-bent functions ; or as a concatenation of four Boolean functions , all of which are simultaneously bent, semi-bent, or 5-valued spectra-functions. In this context, it is essential to ask: When does a bent concatenation (not) belong to the completed Maiorana-McFarland class ? In this article, we answer this question completely by providing a full characterization of the structure of -subspaces for the concatenation of the form and , which allows us to specify the necessary and sufficient conditions so that is outside . Based on these conditions, we propose several explicit design methods of specifying bent functions outside in the special case when , where and are bent functions.
Paper Structure (7 sections, 14 theorems, 13 equations)

This paper contains 7 sections, 14 theorems, 13 equations.

Table of Contents

  1. Preliminaries
  2. Concatenation of two Functions
  3. Concatenation of four Functions
  4. An Application: Designing bent functions outside ${\mathcal{M}}^\#$ of the form $g||h||g||(h+1)$
  5. The necessary and sufficient condition for $f=g||h||g||(h+1)$ to be outside $\mathcal{M}^\#$
  6. A special case of relating $g$ and $h$ in a linear manner
  7. Applying suitable affine transforms

Key Result

Theorem 2

Let $f_1,f_2 \in \mathcal{B}_{n}$ and let $k \in \{ 1, \dots ,n \}$. The function $f=f_1 \vert \vert f_2 \in \mathcal{B}_{n+1}$ has no $(k+1)$-dimensional ${\mathcal{M}}$-subspaces if and only if the following conditions hold:

Theorems & Definitions (30)

  • Definition 1
  • Theorem 2
  • proof
  • Corollary 3
  • Theorem 4
  • proof
  • Remark 5
  • Theorem 6
  • proof
  • Remark 7
  • ...and 20 more