<p>Since the introduction of bent partitions by Anbar and Meidl (Des Codes Cryptogr 90:1081–1101, 2022), several constructions have been presented. All of them correspond to vectorial dual-bent functions with certain conditions, referred to as Condition A, and to partitions of a vector space into partial difference sets, respectively, to Latin square partial difference set packings (LP-packings). We provide the first constructions of bent partitions that do not induce partial difference sets, which shows that bent partitions and LP-packings are inequivalent concepts. We comprehensively analyse vectorial dual-bent functions in the extended Maiorana–McFarland class, which induce bent partitions and corresponding LP-packings. For two classes of bent partitions in characteristic 2, we show that every vectorial bent function from <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {F}_{2^m}\times \mathbb {F}_{2^m}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mn>2</mn> <mi>m</mi> </msup> </msub> <mo>×</mo> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mn>2</mn> <mi>m</mi> </msup> </msub> </mrow> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {F}_{2^k}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mn>2</mn> <mi>k</mi> </msup> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$k < m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo><</mo> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation>, obtained from the bent partitions, can be lifted to a vectorial bent function mapping into <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {F}_{2^m}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mn>2</mn> <mi>m</mi> </msup> </msub> </math></EquationSource> </InlineEquation>. We finally analyse two classes of vectorial bent function constructions.</p>

Analysis of some classes of bent partitions and vectorial bent functions

期刊论文
发表日期：

2026年4月4日

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Nurdagül Anbar,
Fang-Wei Fu,
Tekgül Kalaycı,
Wilfried Meidl,
Jiaxin Wang,
Yadi Wei

摘要

Since the introduction of bent partitions by Anbar and Meidl (Des Codes Cryptogr 90:1081–1101, 2022), several constructions have been presented. All of them correspond to vectorial dual-bent functions with certain conditions, referred to as Condition A, and to partitions of a vector space into partial difference sets, respectively, to Latin square partial difference set packings (LP-packings). We provide the first constructions of bent partitions that do not induce partial difference sets, which shows that bent partitions and LP-packings are inequivalent concepts. We comprehensively analyse vectorial dual-bent functions in the extended Maiorana–McFarland class, which induce bent partitions and corresponding LP-packings. For two classes of bent partitions in characteristic 2, we show that every vectorial bent function from $\mathbb {F}_{2^m}\times \mathbb {F}_{2^m}$ $F_{2^{m}} \times F_{2^{m}}$ to $\mathbb {F}_{2^k}$ $F_{2^{k}}$ , $k < m$ $k < m$ , obtained from the bent partitions, can be lifted to a vectorial bent function mapping into $\mathbb {F}_{2^m}$ $F_{2^{m}}$ . We finally analyse two classes of vectorial bent function constructions.