• DOI:10.1007/s10623-022-01111-6
• Corpus ID: 246634477

Gold functions and switched cube functions are not 0-extendable in dimension n > 5

@article{Beierle2022GoldFA,
  title={Gold functions and switched cube functions are not 0-extendable in dimension n > 5},
  author={Christof Beierle and Claude Carlet},
  journal={Designs, Codes and Cryptography},
  year={2022}
}

• Christof Beierle, C. Carlet
• Published 25 January 2022
• Art
• Designs, Codes and Cryptography

<jats:p>In the independent works by Kalgin and Idrisova and by Beierle, Leander and Perrin, it was observed that the Gold APN functions over <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathbb {F}_{2^5}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>F</mml:mi>
<mml:msup>
<mml:mn>2</mml:mn>
<mml:mn>5</mml:mn>
</mml:msup>
</mml:msub>
</mml:math></jats:alternatives></jats:inline-formula> give rise to a quadratic APN function in dimension 6 having maximum possible linearity of <jats:inline-formula><jats:alternatives><jats:tex-math>$$2^5$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mn>2</mml:mn>
<mml:mn>5</mml:mn>
</mml:msup>
</mml:math></jats:alternatives></jats:inline-formula> (that is, minimum possible nonlinearity <jats:inline-formula><jats:alternatives><jats:tex-math>$$2^4$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mn>2</mml:mn>
<mml:mn>4</mml:mn>
</mml:msup>
</mml:math></jats:alternatives></jats:inline-formula>). In this article, we show that the case of <jats:inline-formula><jats:alternatives><jats:tex-math>$$n \le 5$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mo>≤</mml:mo>
<mml:mn>5</mml:mn>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula> is quite special in the sense that Gold APN functions in dimension <jats:inline-formula><jats:alternatives><jats:tex-math>$$n>5$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mo>></mml:mo>
<mml:mn>5</mml:mn>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula> cannot be extended to quadratic APN functions in dimension <jats:inline-formula><jats:alternatives><jats:tex-math>$$n+1$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mo>+</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula> having maximum possible linearity. In the second part of this work, we show that this is also the case for APN functions of the form <jats:inline-formula><jats:alternatives><jats:tex-math>$$x \mapsto x^3 + \mu (x)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>x</mml:mi>
<mml:mo>↦</mml:mo>
<mml:msup>
<mml:mi>x</mml:mi>
<mml:mn>3</mml:mn>
</mml:msup>
<mml:mo>+</mml:mo>
<mml:mi>μ</mml:mi>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>x</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula> with <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mu $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>μ</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula> being a quadratic Boolean function.</jats:p>