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On the nonlinearity of S-boxes and linear codes

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Abstract

For multi-output Boolean functions (also called S-boxes), various measures of nonlinearity have been widely discussed in the literature but many problems are left open in this topic. The purpose of this paper is to present a new approach to estimating the nonlinearity of S-boxes. A more fine-grained view on the notion of nonlinearity of S-boxes is presented and new connections to some linear codes are established. More precisely, we mainly study the nonlinearity indicator (denoted by \(\mathcal {N}_{\mathrm {v}}\)) for S-boxes from a coding theory point of view. Such a cryptographic parameter \(\mathcal {N}_{\mathrm {v}}\) is more related to best affine approximation attacks on stream ciphers. We establish a direct link between \(\mathcal {N}_{\mathrm {v}}\) and the minimum distance of the corresponding linear code. We exploit that connection to derive the first general lower bounds on \(\mathcal {N}_{\mathrm {v}}\) of non-affine functions from \(\mathbb {F}_{2^{n}}\) to \(\mathbb {F}_{2^{m}}\) for m dividing n. Furthermore, we show that \(\mathcal {N}_{\mathrm {v}}\) can be determined directly by the weight distribution of the corresponding linear code.

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Acknowledgments

The authors are grateful to the anonymous referees for their insightful comments and help in improving the technical quality of this paper. This work is supported by the National Key Basic Research Program of China under Grant 2013CB834204.

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Correspondence to Jian Liu.

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Liu, J., Mesnager, S. & Chen, L. On the nonlinearity of S-boxes and linear codes. Cryptogr. Commun. 9, 345–361 (2017). https://doi.org/10.1007/s12095-015-0176-z

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  • DOI: https://doi.org/10.1007/s12095-015-0176-z

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