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<p>In this paper, we study the Hamming distance between vectorial Boolean functions and affine functions. This parameter is known to be related to the nonlinearity and differential uniformity of vectorial functions, while its calculation is, in general, difficult. In 2017, Liu, Mesnager and Chen conjectured an upper bound for this metric. We prove this bound for two classes of vectorial bent functions, obtained from finite quasigroups in characteristic two, and we improve the known bounds for two classes of monomial functions of differential uniformity two or four. For many of the known APN functions of dimension at most nine, we compute the exact distance to affine functions.</p>

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On the minimum Hamming distance between vectorial Boolean and affine functions

  • 期刊论文

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  • 发表日期：

    2025年7月10日

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  • Gábor P. Nagy

摘要

In this paper, we study the Hamming distance between vectorial Boolean functions and affine functions. This parameter is known to be related to the nonlinearity and differential uniformity of vectorial functions, while its calculation is, in general, difficult. In 2017, Liu, Mesnager and Chen conjectured an upper bound for this metric. We prove this bound for two classes of vectorial bent functions, obtained from finite quasigroups in characteristic two, and we improve the known bounds for two classes of monomial functions of differential uniformity two or four. For many of the known APN functions of dimension at most nine, we compute the exact distance to affine functions.