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<p>APN functions offer optimal resistance to differential attacks and are instrumental in the design of block ciphers in cryptography. While finding APN functions is very difficult in general, a promising way to construct APN functions is through symmetric matrices called Quadratic APN matrices (QAM). It is known that the search space for the QAM method can be reduced by means of orbit partitions induced by linear equivalences. This paper builds upon and improves these approaches in the case of homogeneous quadratic functions over <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {F}_{2^n}\)</EquationSource> </InlineEquation> with coefficients in the subfield <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {F}_{2^m}\)</EquationSource> </InlineEquation>. We propose an innovative approach for computing orbit partitions for cases where it is infeasible due to the large search space, resulting in the applications for the dimensions <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\((n,m)=(8,4)\)</EquationSource> </InlineEquation>, and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\((n,m)=(9,3)\)</EquationSource> </InlineEquation>. We find and classify, up to CCZ-equivalence, all quadratic APN functions for the cases of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\((n,m)=(8,2)\)</EquationSource> </InlineEquation>, and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\((n,m)=(10,1)\)</EquationSource> </InlineEquation>. Also, we show that exhaustive searches for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\((n,m) = (10,2)\)</EquationSource> </InlineEquation>, and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\((n,m)=(8,4)\)</EquationSource> </InlineEquation> are infeasible for the QAM method using currently available means, following partial searches for these cases.</p>

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Pushing the QAM method for finding APN functions further

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  • 发表日期：

    2025年11月26日

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  • Nadiia Ichanska,
  • Simon Berg,
  • Nikolay Kaleyski,
  • Yuyin Yu

摘要

APN functions offer optimal resistance to differential attacks and are instrumental in the design of block ciphers in cryptography. While finding APN functions is very difficult in general, a promising way to construct APN functions is through symmetric matrices called Quadratic APN matrices (QAM). It is known that the search space for the QAM method can be reduced by means of orbit partitions induced by linear equivalences. This paper builds upon and improves these approaches in the case of homogeneous quadratic functions over \(\mathbb {F}_{2^n}\) with coefficients in the subfield \(\mathbb {F}_{2^m}\) . We propose an innovative approach for computing orbit partitions for cases where it is infeasible due to the large search space, resulting in the applications for the dimensions \((n,m)=(8,4)\) , and \((n,m)=(9,3)\) . We find and classify, up to CCZ-equivalence, all quadratic APN functions for the cases of \((n,m)=(8,2)\) , and \((n,m)=(10,1)\) . Also, we show that exhaustive searches for \((n,m) = (10,2)\) , and \((n,m)=(8,4)\) are infeasible for the QAM method using currently available means, following partial searches for these cases.