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<p>Rotation symmetric Boolean functions (RSBFs) have received considerable attention for their excellent cryptographic properties and efficient implementation. These functions have been extensively studied in relation to bentness, correlation immunity, and nonlinearity. While numerous constructions exist for balanced odd-variable RSBFs with optimal algebraic immunity, developing balanced even-variable RSBFs with optimal algebraic immunity remains a significant and largely unresolved challenge. In this paper, we present a novel construction of balanced <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(2^{k}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mi>k</mi> </msup> </math></EquationSource> </InlineEquation>-variable RSBFs with optimal algebraic immunity <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\((k\ge 4)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>≥</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> using integer compositions. The constructed functions obtain a nonlinearity superior to previous construction methods, and their algebraic degree can be maximized under certain conditions.</p>

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Construction of balanced \(2^{k}\)-variable rotation symmetric Boolean functions with optimal algebraic immunity

  • 期刊论文

  • 发表日期：

    2026年7月22日

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  • Jiao Du,
  • Xiaoting Chen,
  • Hui Zhang,
  • Longjiang Qu,
  • Chao Li

摘要

Rotation symmetric Boolean functions (RSBFs) have received considerable attention for their excellent cryptographic properties and efficient implementation. These functions have been extensively studied in relation to bentness, correlation immunity, and nonlinearity. While numerous constructions exist for balanced odd-variable RSBFs with optimal algebraic immunity, developing balanced even-variable RSBFs with optimal algebraic immunity remains a significant and largely unresolved challenge. In this paper, we present a novel construction of balanced \(2^{k}\) 2 k -variable RSBFs with optimal algebraic immunity \((k\ge 4)\) ( k ≥ 4 ) using integer compositions. The constructed functions obtain a nonlinearity superior to previous construction methods, and their algebraic degree can be maximized under certain conditions.