<p>The construction of Boolean functions with good cryptographic properties over subsets of vectors with fixed Hamming weight is significant for lightweight stream ciphers like FLIP. This article introduces a general construction for a class of Weightwise Almost Perfectly Balanced (WAPB) Boolean functions, based on the action of a cyclic permutation group on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {F}_2^n$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="double-struck">F</mi> <mn>2</mn> <mi>n</mi> </msubsup> </math></EquationSource> </InlineEquation>. This class generalizes the Weightwise Perfectly Balanced (WPB) <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n = 2^m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <msup> <mn>2</mn> <mi>m</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>-variable Boolean function construction by Liu and Mesnager to any <i>n</i>. We establish theoretical bounds on the nonlinearity and weightwise nonlinearity of the resulting functions. Particularly, we explore two significant permutation groups, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\langle \psi \rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <mi>ψ</mi> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\langle \sigma \rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <mi>σ</mi> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation> is a distinct binary-cycle permutation and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> is a rotation. Beyond the theoretical analysis of nonlinearity and weightwise nonlinearity, we perform an extensive experimental study for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$n\le 4\le 20$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≤</mo> <mn>4</mn> <mo>≤</mo> <mn>20</mn> </mrow> </math></EquationSource> </InlineEquation>, examining nonlinearity, weightwise nonlinearity, algebraic immunity, and weightwise algebraic immunity of these classes of functions. The results confirm that the proposed WAPB functions achieve high values across these important cryptographic parameters, demonstrating their practical relevance for cryptographic designs that require balancedness across fixed-weight slices.</p>

Weightwise almost perfectly balanced functions, construction from a permutation group action view

期刊论文
发表日期：

2026年4月9日

查看全文

Deepak Kumar Dalai,
Krishna Mallick,
Pierrick Méaux

摘要

The construction of Boolean functions with good cryptographic properties over subsets of vectors with fixed Hamming weight is significant for lightweight stream ciphers like FLIP. This article introduces a general construction for a class of Weightwise Almost Perfectly Balanced (WAPB) Boolean functions, based on the action of a cyclic permutation group on $\mathbb {F}_2^n$ $F_{2}^{n}$ . This class generalizes the Weightwise Perfectly Balanced (WPB) $n = 2^m$ $n = 2^{m}$ -variable Boolean function construction by Liu and Mesnager to any n. We establish theoretical bounds on the nonlinearity and weightwise nonlinearity of the resulting functions. Particularly, we explore two significant permutation groups, $\langle \psi \rangle $ $⟨ ψ ⟩$ and $\langle \sigma \rangle $ $⟨ σ ⟩$ , where $\psi $ $ψ$ is a distinct binary-cycle permutation and $\sigma $ $σ$ is a rotation. Beyond the theoretical analysis of nonlinearity and weightwise nonlinearity, we perform an extensive experimental study for $n\le 4\le 20$ $n \leq 4 \leq 20$ , examining nonlinearity, weightwise nonlinearity, algebraic immunity, and weightwise algebraic immunity of these classes of functions. The results confirm that the proposed WAPB functions achieve high values across these important cryptographic parameters, demonstrating their practical relevance for cryptographic designs that require balancedness across fixed-weight slices.