<p>A crooked function is a permutation for which each of its differential sets is the complement of a hyperplane. In this paper, we introduce the concept of locally crooked permutations, for which at least one of their differential sets is the complement of a hyperplane. We establish a connection between the notions of locally crooked permutation and complete permutation. Specifically, complete permutations over <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12095_2025_830_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}_{2^{n}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mn>2</mn> <mi>n</mi> </msup> </msub> </math></EquationSource> </InlineEquation> can be obtained from locally crooked permutations over <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12095_2025_830_Article_IEq4.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}_{2^{n+1}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </msub> </math></EquationSource> </InlineEquation>, and vice versa. As concrete results, we construct several complete permutations with best-known differential uniformity and nonlinearity, as well as several locally crooked permutations with differential uniformity of 4 over <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12095_2025_830_Article_IEq5.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}_{2^{2n}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mn>2</mn> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msup> </msub> </math></EquationSource> </InlineEquation>. Furthermore, we observe that the existence of APN permutations that are locally crooked over <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12095_2025_830_Article_IEq6.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}_{2^{2n+2}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mn>2</mn> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> </msub> </math></EquationSource> </InlineEquation> is closely related to the notion of nonlinear complete permutation over <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12095_2025_830_Article_IEq7.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}_{2^{2n+1}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mn>2</mn> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </msub> </math></EquationSource> </InlineEquation>.</p>

The locally crooked permutations and the complete permutations over $\mathbb {F}_{2^{n}}$

期刊论文
发表日期：

2025年7月21日

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Li Shuai,
Miao Li

摘要

A crooked function is a permutation for which each of its differential sets is the complement of a hyperplane. In this paper, we introduce the concept of locally crooked permutations, for which at least one of their differential sets is the complement of a hyperplane. We establish a connection between the notions of locally crooked permutation and complete permutation. Specifically, complete permutations over $\mathbb {F}_{2^{n}}$ $F_{2^{n}}$ can be obtained from locally crooked permutations over $\mathbb {F}_{2^{n+1}}$ $F_{2^{n + 1}}$ , and vice versa. As concrete results, we construct several complete permutations with best-known differential uniformity and nonlinearity, as well as several locally crooked permutations with differential uniformity of 4 over $\mathbb {F}_{2^{2n}}$ $F_{2^{2 n}}$ . Furthermore, we observe that the existence of APN permutations that are locally crooked over $\mathbb {F}_{2^{2n+2}}$ $F_{2^{2 n + 2}}$ is closely related to the notion of nonlinear complete permutation over $\mathbb {F}_{2^{2n+1}}$ $F_{2^{2 n + 1}}$ .

The locally crooked permutations and the complete permutations over \(\mathbb {F}_{2^{n}}\)

摘要