Vectorial Bent Functions With Non-Weakly Regular Components

2024/07/29 by Ayça Çeşmelioğlu, Wilfried Meidl
Computer Science · Biochemistry, Genetics and Molecular Biology · #Coding theory and cryptography #Chemical Synthesis and Analysis

paper · doi:10.1109/tit.2024.3434481

Abstract

It is shown that the generalized Rothaus construction of p-ary bent functions can be extended to a construction of a vectorial bent function with non-weakly regular components, for which in general the duals are not a bent function, i.e., they belong to the class of non-dual bent functions. This complements results on other two constructions of non-weakly regular bent functions, the generalized Maiorana-McFarland construction and the semi-direct sum, for which vectorial versions are presented and the properties of their duals are investigated in the literature. The distribution of the values of the Walsh transform of vectorial bent functions (with non-weakly regular components) is then analysed in detail. Among others, a condition on the values of the Walsh transform of a vectorial bent function from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">\mathbb F_pⁿ </tex-math></inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">\mathbb F_p^m </tex-math></inline-formula> is presented, which implies that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">m ≤ \lceil n/2\rceil </tex-math></inline-formula>. This refines a classical result by Nyberg, which states that for an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">(n,m) </tex-math></inline-formula> bent function, n even, with only regular components, m can be at most <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">n/2 </tex-math></inline-formula>. Some results on the weight distribution of codes obtained from vectorial bent functions with non-weakly regular components complement the article.

Citations

• Linear Codes With Two or Three Weights From Weakly Regular Bent Functions

Related

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• There are infinitely many bent functions for which the dual is not bent
• A Note on "Constructing Bent Functions Outside the Maiorana-McFarland Class Using a General Form of Rothaus"
• A Construction of Weakly and Non-Weakly Regular Bent Functions
• Bent functions using Maiorana-McFarland secondary construction