<p>We prove that for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> the map <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\chi :\mathbb {F}_q^n \rightarrow \mathbb {F}_q^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>χ</mi> <mo>:</mo> <msubsup> <mi mathvariant="double-struck">F</mi> <mi>q</mi> <mi>n</mi> </msubsup> <mo stretchy="false">→</mo> <msubsup> <mi mathvariant="double-struck">F</mi> <mi>q</mi> <mi>n</mi> </msubsup> </mrow> </math></EquationSource> </InlineEquation>, defined by <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$y=\chi (x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>y</mi> <mo>=</mo> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$y_i = x_i + x_{i+2}\cdot (1+x_{i+1})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>=</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>·</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$1\le i \le n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>, is bijective if and only if <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$q=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and <i>n</i> is odd, as it was conjectured by Schoone and Daemen in 2024.</p>

On the bijectivity of the map $\chi $

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2025年2月21日

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Anna-Maurin Graner,
Björn Kriepke,
Lucas Krompholz,
Gohar Kyureghyan

摘要

We prove that for $n>1$ $n > 1$ the map $\chi :\mathbb {F}_q^n \rightarrow \mathbb {F}_q^n$ $χ : F_{q}^{n} \to F_{q}^{n}$ , defined by $y=\chi (x)$ $y = χ (x)$ with $y_i = x_i + x_{i+2}\cdot (1+x_{i+1})$ $y_{i} = x_{i} + x_{i + 2} \cdot (1 + x_{i + 1})$ for $1\le i \le n$ $1 \leq i \leq n$ , is bijective if and only if $q=2$ $q = 2$ and n is odd, as it was conjectured by Schoone and Daemen in 2024.

On the bijectivity of the map \(\chi \)

摘要