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<p>The notion of generalized almost perfect nonlinear (GAPN) functions extends the classical concept of almost perfect nonlinear (APN) functions, which are optimal with respect to differential uniformity over finite fields of even characteristic. Since its introduction, the study of GAPN functions has primarily concentrated on monomial and binomial GAPN functions over the finite field <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\mathbb F}_{{p}^{n}}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mrow> <mi mathvariant="double-struck">F</mi> </mrow> </mrow> <mrow> <msup> <mrow> <mi>p</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msup> </mrow> </msub> </math></EquationSource> </InlineEquation>. In this paper, the authors focus on characterizing GAPN functions over <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\mathbb F}_{{p}^{n}}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mrow> <mi mathvariant="double-struck">F</mi> </mrow> </mrow> <mrow> <msup> <mrow> <mi>p</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msup> </mrow> </msub> </math></EquationSource> </InlineEquation> from the Dickson polynomials and reversed Dickson polynomials. Additionally, the authors present one class of GAPN functions derived from known GAPN functions.</p>

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Generalized Almost Perfect Nonlinear Functions from (Reversed) Dickson Polynomials over Finite Fields

  • 期刊论文

  • 发表日期：

    2026年5月8日

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  • Yan-Ping Wang,
  • Yiwen Chen,
  • Weiguo Zhang,
  • Dengguo Feng

摘要

The notion of generalized almost perfect nonlinear (GAPN) functions extends the classical concept of almost perfect nonlinear (APN) functions, which are optimal with respect to differential uniformity over finite fields of even characteristic. Since its introduction, the study of GAPN functions has primarily concentrated on monomial and binomial GAPN functions over the finite field \({\mathbb F}_{{p}^{n}}\) F p n . In this paper, the authors focus on characterizing GAPN functions over \({\mathbb F}_{{p}^{n}}\) F p n from the Dickson polynomials and reversed Dickson polynomials. Additionally, the authors present one class of GAPN functions derived from known GAPN functions.