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On higher order nonlinearities of Boolean functions

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Abstract

When analyzing the security of block ciphers and stream ciphers, the \(r-th\) order nonlinearity of a Boolean function is crucial. They also have a prominent place in coding theory because the \(r-th\) order nonlinearity of Boolean functions is connected to the covering radius of \(\mathcal{R}\mathcal{M}(r,m)\), i.e., Reed-Muller code. In this study, we determine the lower bound for the higher-order nonlinearity of the two classes of Boolean functions listed below.

  1. 1.

    \(f_{\alpha }(u)=tr_1^m(\alpha u^d)\), where d is the Niho exponent constructed by Dobbertin et al. (J. Comb. Theory Ser. A 113:779–798, 2006).

  2. 2.

    \(g_{\alpha }(u)=tr_1^m(\alpha u^d)\), where \(d=2^p-2\). For all \(u\in \mathbb {F}_{2^m},~\alpha \in \mathbb {F}_{2^m}^*\) and \(m=2p\).

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Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology Raipur, G E Road, Raipur, 492010, Chhattisgarh, India

    Sampada Tiwari & Deepmala Sharma

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  1. Sampada Tiwari
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  2. Deepmala Sharma
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Both authors made substantial contributions to the conception. Sampada Tiwari prepared the original draft of the Manuscript. Both authors reviewed the manuscript.

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Correspondence to Deepmala Sharma.

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Tiwari, S., Sharma, D. On higher order nonlinearities of Boolean functions. Cryptogr. Commun. 15, 821–830 (2023). https://doi.org/10.1007/s12095-023-00643-5

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